Nonlinear development of inertial instability in a barotropic shear

Transcription

1 Nonlinear development of inertial instabilit in a barotropic shear Riwal Plougonven Laboratoire de Météorologie Dnamique, ENS, IPSL, 24 rue Lhomond, 755 Paris, France Vladimir Zeitlin Laboratoire de Météorologie Dnamique, UPMC, IPSL, 24 rue Lhomond, 755 Paris, France (Dated: September, 9) Inertial instabilit is investigated numericall in a two-dimensional setting in order to understand its nonlinear stage and saturation. To focus on fundamental mechanisms, a simple barotropic shear U() = tanh on the f-plane is considered. The linear stabilit problem is first solved analticall, and the analtical solutions are used to benchmark numerical simulations. A simple scenario of the nonlinear development of the most unstable mode was recurrentl observed in the case of substantial diffusivit: while reaching finite amplitude the unstable mode spreads laterall, distorting the initiall vertical instabilit one. This proccess produces strong vertical gradients which are subsequentl annihilated b diffusion, making the flow barotropic again but with the shear spread over a wider region. In course of such evolution, unexpectedl, strong negative absolute vorticit anomalies are produced. In weakl diffusive simulations, the horiontal spreading of the unstable motions and the enhancement of the anticclonic vorticit extremum persist, but small-scale motions/instabilities render the flow considerabl more complex. It is known that the barotropic component of the final state can be predicted from the conservation of momentum. Our simulations confirm the relevance of this simple prediction in the cases investigated, regardless of resolution and diffusion. The baroclinic component of the final state is also analed and three tpes of structures are identified: persistent stationar stratification laers, sub-inertial waves trapped in the anticclonic shear, and freel propagating inertia-gravit waves. The sub-inertial waves and the stratification staircase have clear signatures, and can therefore help to identif the regions that have undergone inertial instabilit. PACS numbers: I. INTRODUCTION Smmetric instabilit occurs in rotating fluids when the potential vorticit (PV) is of opposite sign to the Coriolis parameter [], and is the geophsical analog of centrifugal, or Talor-Couette, instabilit. Inertial instabilit is the special case when the relative vorticit is stronger than f in a certain region of the flow, where f is the (local) Coriolis parameter [2]. It occurs in the atmosphere and oceans in the vicinit of the Equator due to either meridional shear [3, 4] or inter-hemisphere advection [5], or due to diabatic processes modifing the PV [2]. Occurences of inertial instabilit due to advection can be found at significant distances from the Equator [6, 7]. Linear aspects of inertial instabilit were extensivel studied, both for onall smmetric flows [4, 8 ], flows with onal variations [2, 3] and oscillating flows at the Equator [4]. For onall smmetric flows in an inviscid fluid, linear theor shows that the growth rates monotonicall increase for increasing vertical wavenumber. However, vertical wavelengths found in observations, for in- Electronic address: riwal.plougonven@poltechnique.org stance in the equatorial stratosphere and mesosphere at solstice, are not consistent with theoretical predictions based solel on mlecular diffusion, making it necessar to understand the nonlinear stage of the instabilit in order to explain the vertical scale selection []. It has been hpothesied that Kelvin-Helmholt (KH) instabilities inhibiting modes with smaller vertical scales pla a role, and this has been tested convicingl with simulations relevant to the equatorial middle atmosphere [6]. In a different configuration the role of Kelvin-Helmholt instabilities in the equilibration of smmertic instabilit has also been highlighted b numerical simulations in the oceanic context [7]. In the diffusive regime, the nonlinear behaviour of inertial instabilit has been described in depth via analtical and numerical approaches for a uniform shear near the Equator []. In the weakl nonlinear regime, it was found that the most unstable mode would grow to a finte amplitude, induce persistent mean-flow changes sufficient to neurtalie the flow (homogeniation of f Q to a small, negative value, with Q the potential vorticit) and then deca. As nonlinearit increases, the overturning cells extend poleward, mixing angular momentum and homogeniing fq over a wider region. It had been noticed previousl [], for an unstratified flow on the f-plane, that a latitudinall narrow mode of inertial instabilit could,

2 through its nonlinear development, homogenie a latitudinall wide region. A similar behaviour has also been documented for axismmetric flows: numerical simulations of inertial instabilit in barotropic vortices have shown the formation of dipoles that move horiontall and mix angular momentum throughout a larger region than the initiall unstable one []. Regarding the end state of evolution of the inertial instabilit, it has been shown recentl that it can be predicted from simple momentum conservation considerations, in what concerns its barotropic component []. The argument is similar to that for geostrophic adjustment of a barotropic jet [2, 22], with the notable difference that potential vorticit is not conserved for fluid parcels where it is initiall negative. Several questions arise, regarding the relevance of this prediction, as well as its sensitivit to the parameters of the numerical simulations, such as resolution and viscosit. Although the end state does not consist onl of the barotropic component, the baroclinic component has received little attention up to now. The construction of the verticall averaged state turns out to be quite simple, et some developed structures with fine vertical scales ma persist. Thorpe and Rotunno [23], in their numerical investigation of the nonlinear development of smmetric instabilit, noted that the instabilit drives the flow toward a state in which regions of negative PV ma still exist, but are broken down to small, fragmented patches, each having insufficient spatial extent for instabilit to develop (at least at the resolution used). Hence several questions arise regarding the end state of evolution of the inertial instabilit: what tpe of baroclinic motions persist, and how intense are the? How sensitive to dissipation and resolution are the? Can the give a clear signature of the former presence of inertial instabilit? For flows with strong shears, it should be kept in mind that barotropic instabilit ma compete with inertial instabilit [, ]. Inertial instabilit will nevertheless have the dominant growth rates for the flows considered here (see [], section 4), and hence the possibilit of barotropic instabilit will not be investigated. The object of the present stud is to investigate the nonlinear development of inertial instabilit, with emphasis on the details of nonlinear saturation and its persistent signatures. Our strateg consists in choosing a configuration simple enough to be analticall solved in the linear approximation. The explicitl known unstable mode will allow us to benchmark the numerical model at the initial growth stage, allowing thus for neat investigation of the nonlinear stage. The paper is organied as follows: the linear stabilit problem is solved for a barotropic shear laer in section II. The numerical simulations of inertial instabilit are described and benchmarked in section III. The nonlinear evolution and saturation of the instabilit are presented in section IV. The question of the final state is addressed in section V, and conclusions are presented in section VI. II. ANALYTICAL SOLUTIONS OF THE LINEAR STABILITY PROBLEM A motivation for considering a barotropic shear on the f-plane is the possibilit to solve analticall the linear stabilit problem. The equations of motion are introduced and scaled in subsection II A, and the linear stabilit problem is solved in subsection II B. The growth rates and structures of the unstable modes are described in subsection II C. A. General form of the eigenvalue problem Following [6], we start from the primitive equations in the Boussinesq approximation and including onl the vertical diffusion, with the tildes denoting dimensional variables: Dũ H D t + f ũ H + H φtot = ν 2 ũ H 2, (a) D w D t + φ tot g θ tot = ν 2 w θ r 2, D θ tot D t H ũ H + w =. (b) = κ 2 θ tot 2, (c) (d) Here ũ H is the horiontal velocit, D/D t = t + ũ is the full Lagrangian derivative, f = f k is the Coriolis parameter times the unit vertical vector, w is the vertical velocit, and ν and κ are vertical (turbulent) viscosit and diffusivit, respectivel. In the atmospheric context, φ tot is the total geopotential, θ tot is the total potential temperature, g is gravit, θ r is a reference potential temperature. The vertical coordinate is the modified pressure coordinate introduced b [24] and for shallow laers it differs onl slighlt from geometric height. In the oceanic context, is geometric height, φ tot is P/ ρ, and θ is densit. In what follows we will consider f >, and negative PV will correspond to inertiall unstable regions. Finall, we suppose that the smallest scales occur in the vertical direction, that is wh onl vertical diffusion is included. We restrict our attention to flows that are smmertic in x: x. From the incompressibilit constraint (d) we can then introduce a streamfunction ψ such that ṽ = ψ and w = ỹ ψ. The potential temperature is split into a background one that varies onl in the vertical and a perturbation: θ tot (ỹ, ) = Θ( )+ θ(ỹ, ). For simplicit we will assume that N 2 = g/θ r (d Θ/d) is constant and that the Prandtl number is unit, i.e. κ = ν. We also introduce the notation χ = g θ/ θ r. We now proceed to scale the equations in order to anale the instabilit of barotropic shear regions. The shear flow U tanh(ỹ/l) provides a natural horiontal lengthscale L and velocit scale U. As the basic flow contains

3 no vertical lengthscale, it is natural to choose H such that the Burger number N 2 H 2 /f 2 L 2 =. In the equations for the non-dimensional (no-tilde) variables (u, ψ, φ, χ), three parameters appear: R = U/fL, the Rossb number, which measures the intensit of the shear relative to planetar vorticit. It appears in the Lagrangian derivative D = t + R( ψ ψ ). γ = ν/fh 2, the Ekman number, which measures the importance of diffusion relative to rotation. Because inertial instabilit develops at the smallest available vertical scales in the inviscid fluid, diffusion will in fact be important even for small values of γ = ν/fh 2. Note that the Renolds number is obtained as Re = R/γ. δ = H/L the aspect ratio. As the Burger number is unit, we also have δ = f/n. For atmospheric and oceanic flows one has δ, and so the hdrostatic approximation (δ ) is justified. In the case of the inviscid fluid (γ = ) it is wellknown (e.g. [24]) that onall smmetric flows ( x ) have two Lagrangian invariants: geostrophic momentum and potential temperature. The PV ma be expressed in terms of them. The non-dimensional expressions of the invariants are respectivel: U() M() PV() M = u R, and Θ = χ + R. (2) R= FIG. : Profile of velocit U() (top), of geostrophic momentum M() (middle), and of potential vorticit Q() = R U (bottom) for the basic state, corresponding to three values of R:.5 (dashed), 2 (thin) and 3 (thick). Also indicated in both figures are the bounds of the unstable regions (du/d > R ) for the three values of R. Now, we consider a background barotropic flow that is in geostrophic equilibrium, U() = Φ b, and hence is a stationar solution of the equations. Lineariing about this flow (u = U + u, φ = Φ b + φ, χ = χ, ψ = ψ ), the R=2 R=3 following equations are obtained: Du + ( RU )ψ =, Dψ u φ =, δ 2 Dψ χ + φ = Dχ + ψ =, (3a) (3b) (3c) (3d) where D ( t γ ) and subscripts denote partial derivatives. Looking for normal mode solutions of the form: ψ(,, t) = ˆψ()e i(m ωt) and combining equations (3) following standard manipulations, a single equation for ˆψ() is obtained: ˆψ + m 2 ˆω 2 δ 2 [ˆω2 ( RU )] ˆψ =, (4) where notation ˆω = ω + i γm 2 is introduced. This equation gives the structure of normal modes within the barotropic shear. It describes a continuous spectrum of inertia-gravit waves that are freel propagating, subinertial waves that are trapped in the anticclonic shear region, and, possibl, unstable modes if the shear is strong enough [22]. B. Explicit solutions for a tanh shear laer Equation (4) was much studied in quantum mechanics in the particle-in-a-potential-well context, e.g. [], and we can profit from this knowledge. The inertial instabilit of a tanh shear profile has previousl been discussed [] using an asmptotic approach which included finite along-stream wavenumbers. The analsis described below ields a compact and explicit form of the dispersion relation in the case with no along-stream variations, both with and without the hdrostatic approximation. We choose the standard representation of the shear flow: U() = tanh(). (5) Note that the velocit- and the length-scales being alread fixed, no parameter enters this definition. The profiles of M and Θ will change with R, as follows from equation (2) and Figure. Injecting (5) into equation (4) ields: ˆψ + m 2 ˆω 2 δ 2 [ ( ˆω 2 R )] ˆψ cosh 2 =. (6) Following [], we change variables as ξ = tanh(), and transform this equation into [ d ( ξ 2 ) d ] ] ˆψ + [s(s + ) ǫ2 ˆψ dξ dξ ξ 2 =, (7a) where s(s + ) = m2 R ˆω 2 δ 2, ǫ2 = m2 ( ˆω 2 ) ˆω δ 2. (7b)

4 ω 2 σ.5.5 Freel propagating waves Trapped waves Unstable modes n=4 n=3 n=2 n= n= m γ=.3. γ= m FIG. 2: Upper panel: Dispersion curves ˆω 2 (m) for R =.5 with δ = 2 for n= to 4. Two sets of curves are shown: from the full expression () in thin black lines, and from the hdrostatic expression () in thick, dashed, gre lines. Horiontal lines indicate the separation (thin line, ˆω 2 = ) between the continuous spectrum of freel propagating waves and discrete spectrum of trapped waves, the separation (thick line, ˆω 2 = ) between trapped waves and unstable modes, and the asmptote for the lowest value of ˆω 2 (thin line, ˆω 2 = R =.5). Lower panel: Growth rate σ as a function of the vertical wavenumber m, for the fundamental mode n =. The three curves correspond to γ =, 3. 3 and Maximum growth rates are found on the thick gra curve as γ varies, tending for γ towards the asmptote shown as a thin black line. Note that we have supposed that ˆω 2 < to define ǫ. Solutions that are finite for ξ = ( + ) are of the form: ˆψ = ( ξ 2 ) ǫ/2 F [ǫ s, ǫ + s + ; ǫ + ; 2 ] ( ξ), (8) where F is the hpergeometric function ([26], chapter ). For ψ to remain bounded at ξ = ( ), we need to have ǫ s = n, with n =,, 2, 3... (9) The parameter n corresponds to the number of nodes of ψ(): inside the potential well, where ˆω 2 ( RU ) >, ψ oscillates and has n nodes. Outside the potential well the function ψ() decas exponentiall (e.g. Fig. 4 in [22]). The constraint (9), together with definitions (7b), ields the following dispersion relation: ˆω G(ˆω 2 2, m, n, δ, R) =2m δ 2ˆω 2 () + 2n + γ= + 4m2 R δ 2ˆω 2 =. For the configurations with δ under consideration it turns out that this form is redundant. The hdrostatic limit (δ = ) of the dispersion relation can be used instead: ˆω 2 = ( 4m 2 2n + ) 2 + 4Rm 2, () which is valid onl for ˆω 2 < (cf eq. (7b)), i.e. onl for n(n+) m > R. The latter condition implies that there is alwas an interval of wavenumbers m close to ero that are stable. The width of this interval increases with n (Fig. 2). The upper panel of Figure 2 describes ˆω 2 obtained both b the hdrostatic () and non-hdrostatic () relations for a Rossb number R =.5 and δ = 2. The curves are indistinguishable, which confirms that for the low values of δ, commonl found in the atmosphere, the hdrostatic limit is largel sufficient. We will use the form () in what follows. C. Growth rates of the smmetric unstable modes It is insightful to first examine the dispersion relation () in the inviscid case, i.e. when ˆω = ω. For strongl anticclonic flows (R > ), ˆω 2 becomes negative for sufficientl large m (the corresponding mode then changes from a trapped sub-inertial wave [27] to an inertiall unstable mode). Figure 2 shows that, as m increases, the n = mode is the first to become unstable, and it is followed b modes with increasing n. The growth rates increase monotonicall with m and tend toward σ max = R as m for all values of n, though more slowl as n increases. The inclusion of diffusion in the vertical onl modifies the definition of ˆω = ω + i γm 2. The growth rates of unstable inviscid modes diminish, and for large enough values of m these modes become decaing. Hence, as illustrated in the lower panel of Fig. 2, onl a finite range of vertical wavenumbers m are unstable, and the maximum growth rate is attained at a finite value m max, in contrast to the inviscid case (well-defined scalings for the variation of m max as γ can be obtained [28, 29]). This fact will allow for well-resolved numerical simulations in what follows. The linear theor provides not onl the growth rates, but also the structure of the modes. The streamfunction is ψ(,, t) = ˆψ()e i(m ωt), with ˆψ given b (8). Other variables are expressed in terms of ˆψ as follows: û = mˆω ( RU ) ˆψ, ˆv = im ˆψ, (2a) ŵ = d ˆψ d, d ˆχ = iˆω ˆψ d. (2b) The unstable modes are confined within the region with negative PV. Outside the region of anticclonic vorticit, the deca exponentiall. Within the unstable region,

5 the most unstable mode (n = ) consists of a stack of overturning cells [4], as illustrated in the upper panel of Figure 5. III. LINEAR INSTABILITY IN NUMERICAL SIMULATIONS The analtical solutions described above provide a benchmark for validation of the linear stage of the instabilit in the numerical simulations. The numerical setup of the simulations is described in subsection III A, and the comparison with the analtical solutions in subsection III B. A. Numerical setup and simulation parameters The simulations were carried out with the Weather Research and Forecast model (WRF, version 2.2, [3]), which allows both idealied simulations and real-case studies for atmospheric flows. Chosing a communit model that has a wide range of capabilities and is widel used has several advantages: it makes it easier for other investigators to reproduce present results, and what is more important, it opens the possibilit, in further studies using the same model, to increase the complexit and realism of simulations (3 dimensions, incorporation of moist processes) and to move toward case studies which can be compared with observations (e.g. for air masses advected from one hemisphere to the other, as described b [5, 6]). The model integrates the equations for a full compressible, nonhdrostatic atmosphere in flux form with terrain-following hdrostatic pressure as the vertical coordinate [3]. The prognostic variables are the velocit components, the potential temperature, geopotential and the surface pressure, and the are discretied using a staggered Arakawa C-grid. The equations are integrated in time using a time-split third order Runge-Kutta scheme with a smaller time step for acoustic modes. The top of the model is a constant pressure surface. One implication of using WRF is that the simulated flows will differ in several aspects from the flows for which analtical solutions were obtained: the model describes the evolution of a compressible atmosphere, whereas the Boussinesq approximation was used in section II. The fluid is bounded below b a rigid, flat bottom with freeslip conditions, and the upper boundar condition corresponds to a free surface with constant pressure. Lateral walls with free-slip boundar conditions close the domain in the transverse direction. Finall, in order to provide dimensionful variables for the code, the horiontal lengthscale L = km for the width of the shear one was chosen. The domain used for the simulations has dimensional width km and a height of about km (the last model level is an isobaric surface, its geometric height is not constant). The dimensional Coriolis parameter is f = 4 s, so that one non-dimensional time unit will correspond to /f = 2.78 h dimensionall. Preliminar simulations showed that the top and bottom boundar conditions introduced undesired effects if the barotropic shear was taken to extend throughout the whole computational domain. Indeed, the unstable mode would not grow homogeneousl in the vertical but would be strongl enhanced near the top boundar. This is presumabl due to the free surface condition at the top boundar, which allows fast surface modes to exist and propagate with a strong signature near the top of the domain. In consequence, to avoid an influence of the top and bottom boundaries, the basic state was chosen to be a barotropic unstable shear in the center of the domain, decaing and becoming stable close to the top and bottom boundaries. B the thermal wind balance, this introduces horiontal gradients of temperature in the region where there is vertical shear. In order to limit the extent of these regions, the basic velocit was chosen to have a slow horiontal decrease far from the barotropic shear region: U(, ) = tanh()(/4) (3) [tanh(6 (/L Z.)) tanh(6 (/L Z.85))] [tanh(2 (/L H.)) tanh(2 (/L H.875))], where L H and L Z are the domain width and height. The resulting distribution of wind and potential temperature is displaed in Figure 3. In the center of the domain, for < < and for < <, the flow is a barotropic shear with U() = tanh() as described in section II, and the boundaries no longer affect the development of the instabilit, although the will pla a role at late times due to the reflection of emitted gravit waves (see Section V B 3) FIG. 3: Contours of the onal velocit U(, ) of the basic state (black lines, dashed for negative values, c.i. of.2) and potential temperature (thin gra lines) for simulations with R = 3. Horiontal axis is, nondimensionalied b L = km, and vertical axis is, nondimensionalied b fl/n = m. Also shown is the ero contour of PV (thick black line), and the region in which the flow is well approximated as a barotropic shear (dashed line). The initial fields for velocit and temperature were obtained numericall, b solving iterativel for the pressure so that the geostrophic onal wind corresponds to

6 R Ekman number γ Resolution.5 : γ = : : γ = : : γ = : : γ = : 6 6 (5: 52 52) TABLE I: Summar of the various parameter values that were used in the 2-dimensional simulations. Numerical growth rate the one prescribed, and that the stratification be such that δ = 2 on average. Wind was deduced from geostrophic equilibrium and potential temperature was obtained from hdrostatic balance. More than fift simulations were carried out, varing the Rossb number R, the diffusion γ and resolution as described in table I. Corresponding Renolds numbers range from to in the simulations where explicit diffusion was included. Simulations will be referenced below b the values of these numbers: for example, simulation R2 γ r3 corresponds to a Rossb number R = 2, a diffusion parameter γ = 3. 3 simulated at resolution For each simulation, a breeding procedure was used to isolate the most unstable growing mode: a simulation is started with a small random perturbation in the temperature field. After 4.3 non-dimensional time units (2 hours dimensionall), the anomal relative to the initial state is isolated, rescaled to a small amplitude and added to the basic state to start a new simulation. This ccle is repeated 5 times. A simulation was then restarted in 52 non-dimensional time units with this mode present in the initial condition with a weak amplitude Analtical growth rate FIG. 4: Comparison of the analtical and numericall obtained growth rates. Circles correspond to runs with R =.5, squares for R = 2, and triangles for R = 3. Smbols are in black for resolution 4 runs, in gra for resolution 3, and in light gra for resolution 2. For a given set of smbols, the analtical growth rate increases as the Ekman number decreases from γ = , to γ =.55 2, to γ = 3. 3 and finall. For R =.5, flows with γ = were stable. velocit for simulation R3 γ3 r4. Both the analtical and the numericall obtained normal modes are shown. The simulated normal mode has a slightl smaller vertical wavelength, but the two structures are otherwise essentiall the same. Analtical.5 B. Comparison with analtical predictions.5.5 The availabilit of analtical solutions for the linear stabilit problem provides a benchmark for the numerical simulations. The growth rates and the structure of the modes obtained numericall are found to agree well with those obtained analticall for configurations which were sufficientl diffusive (Figure 4). If one takes into account the finite resolution while calculating the theoretical maximum growth rate, the agreement improves for the simulations with γ =. Finall, the agreement is not as good for the strongl unstable case (R = 3), in part because the breeding procedure was less successful in isolating the unstable mode in a clean wa (the breeding ccles were too long to ensure that onl linear growth occured) and in part because of insufficient resolution (theoreticall, one would expect modes close to the gridscale). The structure of the numericall obtained modes is also in agreement with that predicted theoreticall. It is illustrated in Fig. 5 b the structure of the meridional Numerical FIG. 5: Meridional (thick, black) and vertical (thin, gra) velocit isolines for the most unstable normal mode of a barotropic shear with R = 3 and γ = The analtical solution (top panel) compares ver well with the numericall obtained normal mode (bottom panel). Contours correspond to the same values on both panels (±/6 [, 2, 3, 4, 5] of the maximum value of v and w, with negative contours dashed), and the bounds of the unstable region are shown (vertical lines).

7 IV. NONLINEAR SATURATION.23 The linear normal mode solution describes an overturning motion (Fig. 5), and uninterrupted growth of these motions will lead to static instabilit in a finite time. For sufficientl diffusive flows, however, it is known that a purel diffusive equilibration ma occur, as described for the case of a constant shear at the Equator in []. A simple scenario of nonlinear saturation of the instabilit can be identified in strongl diffusive simulations (section IV A). Essential features of this scenario persist in weakl diffusive simulations as well, although small-scale motions become much more active (section IV B). Sensitivit to the numerical setup is discussed in section IVC A. Strongl diffusive simulations For strongl diffusive flows, it is expected from previous studies that the instabilit will grow and then deca, leaving behind a modified mean flow []. For strongl unstable situations, the unstable mode is expected to deform and affect a region wider than the initiall unstable region [, ]. We focus below on two specific signatures of this strongl nonlinear case: nonlinearit manifests itself in the horiontal spreading of the unstable motions, and in the temporar enhancement of the maximum anticclonic vorticit. Figure 6 depicts the evolution of the unstable mode in the strongl diffusive, and strongl unstable simulation R3-γ3-r4. The structure of the mode remains smooth and well-resolved, and the mode spreads horiontall once it has reached finite amplitude. This can be understood as a consequence of the modification of the background stratification (weaker stratification ahead of the transverse flow, see Fig. 6, favors spreading). The resulting alternating northward and southward jets do not induce an overall mass flux, but do induce a negative flux of geostrophic momentum. During this process of horiontal spreading, a somewhat counterintuitive effect occurs ahead of the transverse jets. In such regions, the fluid undergoes considerable vertical stretching, as can be seen in the middle column of Fig. 6. For parcels which are initiall unstable, absolute vorticit initiall is negative, and the stretching therefore amplifies the anticclonic vorticit. This is illustrated in Fig. 7 b horiontal profiles of the relative vorticit taken at the center of a northward and a southward jet. Hence, although the instabilit is caused b the negative absolute vorticit, nonlinear evolution temporaril produces its significant intensification (more than doubling, see Fig. 7). This is reminiscent of the behaviour described b Griffiths [] for f Q, where Q is the verticall averaged vorticit anomalies (paragraph 4.6 and Fig. 7 of []). However, the equatorial problem has an asmmetr due to the β-effect such that enhanced negative vorticit anomalies are found onl on the pole FIG. 6: Horiontal spreading of the unstable mode in simulation R3 γ3 r4, for non-dimensional times t=2.9, 4.4, 5.8 and 7.2, from top to bottom. Left: meridional and vertical velocit, as in Fig. 5 but with less contours for clarit (contours are for /3 and 2/3 of max( v ), indicated in the top left). Middle: isolines of potential temperature and of PV (gra shading for positive PV, and contours for PV= (thick) and PV=. (thin). Right: velocit (shading) and PV= contour (white line). Also shown in the lower-right panel are the heights corresponding to the profiles of relative vorticit presented in Fig FIG. 7: Vertical component of the relative vorticit at height =. (plain) and =.4 (dashed) for simulation R3 γ3 r4 (as in Fig. 6, as a function of, at four successive non-dimensional times: t=2.2, 4.4, 6.6 and 8.8, from top to bottom. ward side of the unstable region. In contrast, the vorticit anomalies described above appear on both sides of the initiall unstable region, at different heights. Finall, the saturation of the instabilit can be understood as follows: as the mode spreads out, the column of negative PV becomes severel distorted, in such a wa that meanders offer strong vertical gradients for diffusion to act upon (middle and right columns of Fig. 6). The t=2.2 t=4.4 t=6.6 t=8.8

8 .3 Minimum relative vorticit in the Ro=2 simulations Minimum relative vorticit in the Ro=3 simulations Relative vorticit lower than f in the Ro=3 simulations FIG. 8: Same as Figure 6, but for simulation R2 γ r4. distribution of onal velocit then returns to a barotropic configuration, with the anticclonic shear spread out over a wider region (right column in Fig. 6) so that the flow is stabilied (the width of this region in the final state is discussed in section V). B. Weakl diffusive simulations The strongl diffusive simulations reveal a clear and simple picture of the nonlinear development of the instabilit. For simulations with weaker diffusion, small-scale instabilities come into pla and make the structures more complicated (Fig. 8). No special parameteriation is used in the model to deal with Kelvin-Helmholt instabilities, and it is numerical diffusion which takes care of them, in contrast to the simulations of [6]. Nevertheless, two of the main features outlined above persist and appear to be generic: the horiontal spreading and the increase in the minimum relative vorticit are found sstematicall in the simulations. The nonlinear spreading is found in all simulations, regardless of diffusion and resolution, and will be explained b the tendenc of the fluid towards a neutrall stable end state in section V. The evolution of the minimum of the relative vorticit is quantitativel comparable in all the high-resolution simulations of the strongl unstable shear (R = 3, middle panel of Fig. 9). However, at lower resolutions the localied enhancement of relative vorticit nearl disappears, emphasiing the importance of resolution for this effect. This sensitivit to resolution is not surprising for a quantit that is a spatial derivative. When the shear is weakl unstable (R =.5 or R = 2 with strong diffusion) the enhancement of anticclonic vorticit is reduced because the horiontal spreading becomes too slow relative to diffusion (upper panel of Fig. 9) Time FIG. 9: Minimum of the relative vorticit u in the domain < < and 7 < < 2 for R = 2 (upper panel) and for R = 3 (middle panel) and percentage of points for which the absolute vorticit R u is negative (lower panel) as a function of time. Results are shown for simulations at resolution 4 (black lines) for γ (thick), γ (thick, dashed), γ2 (thin) and γ3 (thin, dashed). Also shown are the results for γ at resolution 3 (gra, thick) and 2 (gra, thin). Yet, there are significant differences between weakl and strongl diffusive simulations. First, some regions of the flow become sucseptible to small-scale KH instabilities (at small Richardson numbers), as well as static (convective) instabilit when R = 3, see Fig.. In consequence, the flow becomes more complex, and the evolution of the region with PV< differs. The initiall unstable region is broken up into smaller and smaller pockets of negative PV (see Fig. 8), and regions with negative anticclonic vorticit persist for much longer time than in the diffusive simulations (lower panel of Fig. 9). Hence, in strong contrast to the more diffusive simulations, well-identifiable small-scale structures persist even once the barotropic component of the flow has reached a stable state, with remaining patches of negative PV and negative absolute vorticit, as described in [23] (Fig. 8 and the lower panel of 9). Long-lasting small-scale structures are further discussed in section V. A few remarks are in order regarding the behavior of the model when no explicit diffusion is included, i.e. when γ =. First, simulations with γ = and γ = 3. 3 are quite comparable for the development and saturation of the instabilit (see Fig. ), which gives us an indication regarding the level of numerical diffusion in the model. Figure also shows that the simulations are much more sensitive to R than to γ, and in consequence that the use of these two parameters separatel has advantages relative to the use of the Renolds number Re = R/γ. Second, we calculated the evolution of the total energ of the flow for simulations with γ =, as conservation of energ is not expected for flows involving instabilities down to the grid scale. From theoretical arguments we can predict what portion of kinetic energ, E K, is expected to be removed from the barotropic

9 component of the flow b the instabilit (see Section V). In simulations with γ =, we found that the kinetic energ loss agreed ver well with this prediction, and that the loss of total energ was comparable to or smaller than E K. For example, for γ = and R = 3 the total energ loss was equal to 8% of E K, i.e. that % of the kinetic energ loss had been converted to potential energ (see Section VB). Hence, although we do not control the numerical diffusion, simulations with γ = exhibit a satisfactor energetics, and are hence useful to provide insights regarding the instabilit and its signature in weakl diffusive flows (the disadvantage of simulations with γ = 3. 3 being that the vertical diffusion, although weak, affects the vertical stratification and damps the small-scale features that persist after the instabilit has saturated). 5 Ri<. in the Ro=2 simulations Ri<. in the Ro=3 simulations Static instabilit in the Ro=3 simulations 5 3 Time FIG. : Percentage of the volume of fluid in the domain < < and 7 < < 2 for which the Richardson number is lower than /4 (upper panel, for simulations with R = 2 and middle panel, for simulations with R = 3) and for which there is static instabilit (lower panel, for R = 3). Lines are as in Fig. 9. C. Sensitivit to other effects To investigate the sensitivit of the simulations to other effects, numerical experiments were carried out with the hdrostatic approximation, with a third dimension, and with higher resolution. For all simulations described above, the model was used in non-hdrostatic mode. The importance of nonhdrostatic effects was tested b switching on the hdrostatic option in WRF, and revealed onl a little difference as could be expected for the parameter regime considered (δ = 2 ). The linear growth rates of the modes were barel changed. The end states were ver similar: in strongl diffusive simulations the differences in velocities were tpicall smaller than %. In weakl diffusive simulations, final states were comparable, with smallscale fluctuations having similar amplitudes, lengthscales and overall structure, though individual details did not match. As a crude preliminar investigation of the potential modifications arising from three-dimensional effects, simulations were also carried out in a three-dimensional setting, with points in the x direction, at resolution r3. The initial state was perturbed with random noise in the potential temperature. It turns out that the resolution used is insufficient to observe an changes in the development of the instabilit. The rms of fluctuations in the x direction remains stationar during the simulation, and differences in the end states are minor and small-scale (e.g. for u, less than 4% for γ, and decreasing further as diffusion increases (less than 2% for γ, less than.4% for γ2). These simulations indicate that if three-dimensional effects modif the development of the instabilit as suggested in [6], higher resolution is necessar to capture them. In fact, in order to investigate these three-dimensional effects and to describe the cascade of instabilities involved would require not onl multipling the sie of our domains b the number of points in the third dimension, e.g., but also increasing the resolution so that Kelvin-Helmholt instabilities, which are non-hdrostatic, can be described. A dramatic increase in resolution (factor /δ) and hence of computing resources, or a different strateg (smaller values of δ) will be necessar to investigate rigorousl these threedimensional effects. Finall, two simulations (R = 3, γ and γ) were also carried out at a higher resolution (r5) to investigate further the sensitivit to resolution. The results were similar to the corresponding simulations at resolution 4, et with some additional small-scale details (see Fig. and 6). V. THE FINAL STATE The previous sections have focused on the development of inertial instabilit in its linear (section III) and nonlinear phase (section IV). The next question is to examine the end state toward which the fluid is evolving. Two preliminar remarks need to be made at this stage: first, the flows considered are constrained to be two-dimensional. Geostrophicall balanced flows are then stationar solutions of the full equations and, if stable, are the most natural candidates for the final adjusted state. Stabilit of such states is guaranteed if PV is everwhere positive. Second, the problem is reminiscent of the geostrophic adjustment problem for a twodimensional jet or front [3 34]. The final adjusted state for an arbitrar initial condition with everwhere positive PV can be determined unambiguousl, being entirel defined b the conservation of M and θ [22]. The flows with strong (but stable) anticclonic shears ma contain sub-inertial waves [22, 27] trapped in the anticclonic region, and hence modifing the standard geostrophic adjustment scenario [35]. The present problem ma be viewed in this context as an adjustment towards an unstable state. Naturall, this

10 latter can not be achieved because the instabilit drives the sstem to another, stable, adjusted state which can not be predicted uniquel from the conservation laws, as nonlinear saturation of the instabilit requires dissipation (cf [36] for a similar situation but with another tpe of instabilit). Predictions for the barotropic component of the final state were given in [] (in the case of rectilinear flows), and [] (in the case of axismmetric flow). The are based on conservation of total linear or angular momentum, respectivel. In particular, [] described the instabilit of a mixing laer similar to the flows we consider, but with U() erf(). Hence two questions arise considering the development of inertial instabilit as an adjustment problem: u u Ro=2, res 4, all γ Ro=2, γ=, all res Ro=3, res 4, all γ Ro=3, γ=, all res how close does the barotropic component of the flow come to the adjusted state predicted b [] at the end of our simulations? 2. what are, if an, the baroclinic fluctuations around this adjusted state? Obviousl, the sensitivit to the model parameters (resolution, diffusivit) should be investigated while answering both questions. A. Predictions for the final state We first recall the prediction of the final state, as given b Kloosteriel, Carenvale and Orlandi []. It is natural to assume that the instabilit suppresses the regions of negative PV. For a barotropic jet, the instabilit ma be inferred from the plots of the verticall averaged geostrophic momentum M() = ( 2 ) 2 Md: unstable regions are those where M increases with, and marginall stable regions are those where M is constant in. As explained in [], a tentative final state can be obtained from a Maxwell rule applied to the plots of M (Fig. ). The initial state, Mi () = U() R, is unstable where M i / >. Let M f () describe the verticall averaged final state. It is natural to assume marginal stabilit M f / everwhere and M f () = M i () far from the initiall unstable one. Graphicall, this means that the region of increasing M i in Fig. is replaced b constant M f in a larger region < < 2, such that M i ( ) = M f ( ) and M i ( 2 ) = M f ( 2 ). In general one more constraint is needed to determine the region of uniform M f. This additional constraint is the conservation of momentum: 2 Mf d = 2 Mi ()d. In the antismmetric configurations considered here, the final state is given b: M f () =, for Y < < Y, and (4) Y M f () =M i () for > Y, with R = tanhy, FIG. : Verticall averaged onal velocit, between = 7 and = 23, in the final states (averaged for one inertial period (2π) around t = 3). The initial velocit U() (thick, gra line) and the bounds of the unstable region (thin vertical lines) are shown for reference. Left column shows results from simulations with R = 2, right column for simulations with R = 3. Upper panels show 4 curves corresponding to the different values of γ used, while the lower panels show four curves corresponding to resolutions -4. Lines correspond to increasing γ (or decreasing resolution) in the order: thick, thick dashed, thin and thin dashed. where Y is the positive, non-ero root of M i (Y ) = (see Fig. ). The barotropic component of the simulated flows at late times was compared with this simple prediction. The agreement is remarkable, as shown in plots of onal velocit (Fig. ), and there is surprisingl small sensitivit to resolution and diffusion. Some sensitivit does appear on the plots of the geostrophic momentum in Fig. 2, showing a better agreement as resolution is increased. In particular, in weakl diffusive simulations and at high enough resolution, the flow is modified beond [ Y, Y ], which is consistent with overshoots occuring during the nonlinear spreading of the unstable motions []. Yet, this sensitivit is less important than in [], possibl because the simulations of [] had higher Renolds numbers, and included horiontal diffusion. In order to illustrate the adjustment to the final state, we show in Fig. 3 the profiles of M() at different times for simulation R2 γ r4. The adjustment is ver rapid (of the order of an inertial period) and, interestingl, after the initial adjustment stage M() hardl fluctuates at all. The availabilit of a simple and efficient prediction of the final state allows to calculate analticall the excess kinetic energ to be removed from the barotropic component of the flow b the developing instabilit. This possibilit was discussed for vortices [] and for barotropic shear profiles []. There are good reasons to believe that that the final state constructed with the method above alwas has less energ than the initial state []. In the present case, the energ loss can be obtained from the

11 M Ro=2, γ=, all res Ro=3, γ=, all res FIG. 2: Same as the lower panels of Fig., but for the verticall averaged geostrophic momentum. awa from boundaries), the equilibration of the flow necessaril involved lateral mixing with fluid having positive PV. In cases where the unstable region has a small vertical extent or when the domain boundaries are close to the unstable region, equilibration ma also involve entrainment of fluid with positive PV from above and below [7] or fluxes from the boundaries [23]..4.2 B. Small-scale fluctuations in the final state M FIG. 3: Verticall averaged geostrophic momentum between = 7 and = 23 for the simulation with R2 γ r4, as in Fig. 2, left panel, with the addition of intermediate states (thin lines for t =.,.5,.75,, 2, 3,... 8). profiles of initial and final velocit, u i = U (5), and ū f deduced from (4): (ū2 Y ) E K = i ū 2 f) d = (tanh 2 () 2 d = 2 Y Y R 2 ( R Y 2 ) 3 R 2, () where we have used the facts that Y Y tanh2 () d = 2(Y tanhy ) and tanhy = Y R. For large R, Y R and the latter expression is readil approximated as E K 4 3 R 2. (6) Whereas the barotropic component of the flow has a simple and robust behavior, the baroclinic component of the end state will be much more sensitive to resolution and dissipation. Yet, for large enough Rossb numbers this component is essential. It has hitherto attracted little attention, and will be analed below. The baroclinic fluctuations are small-scale and fall into three categories: permanent laering in stratification, resulting from the mixing during the nonlinear stage of the instabilit (section V B ), sub-inertial oscillations remaining in the anticclonic region (section VB2) and gravit waves freel propagating awa from the unstable region (section VB3) Kloosteriel et al [] compared the energ loss according to their prediction and in numerical simulations for four barotropic shear profiles, and found reasonnable agreement. As shown b Fig. and as discussed below, our simulations confirm the relevance of this prediction for the energ loss b the mean barotropic flow, for moderate and high Renolds numbers. Thus, our simulations suggest that the barotropic, geostrophicall balanced part of the final state is well predicted from simple arguments, and that the adjustment toward this state is achieved, basicall, regardless of resolution or dissipation. Two final remarks are in order. First, this conclusion applies to flows that are indeed unstable (strong enough diffusion ma inhibit the instabilit altogether), and one ma expect a somewhat different scenario for the flows just over the stabilit threshold []: diffusion can inhibit instabilit for regions with weakl negative PV. Hence, in such a case the equilibration of the instabilit does not necessaril lead the fluid to a state with ero PV, but rather a state with a weak, negative PV. Second, in the present case (barotropic state, FIG. 4: Instantaneous intensit of the stratification χ at late times (t=39) of simulations with resolution 4, γ = and R =.5 (top), R = 2 (middle) and R = 3 (bottom). Also shown several isolines of the same field, averaged over 8.6 time units. Vertical lines indicate the bounds ( Y and Y, cf ())of the modified region in the predicted adjusted state, and maximum value of χ are indicated in the upper left of each panel.

12 3 Modifications of the stratification: laering 2. Sub-inertial oscillations A notable feature of the final state is that it retains a wide region (of non-dimensional width 2R for large enough R) of strong anticclonic vorticit. Trapping of sub-inertial waves is possible in such region. The introduce significant local fluctuations as displaed b Hovmo ller plots in Fig. and are at the origin of a significant portion of the signal. The fact that those are indeed sub-inertial trapped waves is confirmed using Fourier transform in time of the meridional velocit at = (Fig. 7), showing fluctuations dominating at frequencies around a third of the inertial frequenc. This is consistent with the oscillations seen in Fig., with tpical non-dimensional periods of about, i.e. close to 3 2π. The importance of sub-inertial waves can be quantified relative to the prediction of EK (). Fig. shows the time evolution of the kinetic energ of the flow. First, v < v <.53 v <.34 5 v < v <.5 3 Time Time 4 5 FIG. : Hovmo ller diagrams of the meridional velocit at =. Left column: R = 2, right column: R = 3. First row: γ = 3. 3, resolution 4. Second row: γ =, resolution 4. Third row: γ =, resolution 3. 3 v <.75 In the simulations with weak diffusion, stabiliation of the flow involves small-scale mixing. Necessaril, in regions where the fluid is mixed, its stratification weakens. In adjacent regions, stronger stratification will be found, and as a result a staircase vertical profile of potential temperature ma be expected. To check this we are looking below for persistent features in stratification χ. Figure 4 shows vertical cross-sections of stratification χ at a late time of weakl diffusive simulations (γ = ). Persistent laers of alternating stronger and weaker stratification are clearl found in the adjusted region (between Y and Y ). In addition, a reflectional asmmetr with respect to the central axis = of the shear region clearl appears in the case of the simulation with R =.5, reminiscent of the antismmetr of the stratification anomalies in the normal mode (Fig. 6). As the Rossb number increases, the laering becomes less regular, but homogeniation of laers becomes more intense, leading to distinct staircase patterns. Also, one can note that at large Rossb numbers the flow becomes modified outside the theoreticall predicted range [ Y, Y ], as was alread mentioned above. Similar indications of laering can be also be found in the simulations of inertial instabilit at the Equator (Fig. 7 of [6]), which included a parameteriation for Kelvin-Helmholt instabilit. Small-scale structures can also be found in the final PV field of the weakl diffusive simulations. Although PV and stratification anomalies are evidentl related, the PV field is more irregular and does not exhibit the same laered structures. This can be explained b considering for example a region of strong stratification, with onl vertical vorticit: depending on the sign of the absolute vorticit, this can ield either a positive or a negative PV anomal. 3 v <.5. 5 Time FIG. 6: Hovmo ller diagram of meridional velocit for a simulation with γ = and R = 3, as in the right, middle panel of Fig., but for resolution 5. the curves confirm that the energ loss in the simulations is full consistent with the theoretical prediction: for all but strongl diffusive simulations, where energ is continuousl dissipated, the ratio of the simulated energ loss to the theorectiall predicted EK tends to -. Second, the show fluctuations around the final value, indicative of the presence of sub-inertial oscillations for which kinetic energ is periodicall converted to potential energ, and vice-versa. To quantif more precisel the energ that remains trapped in sub-inertial oscillations, Fig. shows the time evolution of the baroclinic part of kinetic energ. In the weakl diffusive simulations (γ and γ) for strongl unstable flow (R = 3) the baroclinic part of kinetic energ (mainl due to the sub-inertial oscillations) represents about 5-% of the kinetic energ loss 2 inertial periods after the saturation of the instabilit. This portion then slowl decas as the waves leak energ to freel propagating waves and are dissipated. The trapped waves are sensitive to the Rossb number, to resolution and to diffusion (Fig. and ). The sensitivit to diffusion is most important. Indeed, as the Rossb number increases, both the excess energ and the width of the anticclonic region in the final state increase. For large enough Rossb numbers the excess energ b unit length of anticclonic region tends to a constant.