Asymmetric inertial instability

Asymmetric inertial instability V. Zeitlin Institut Universitaire de France 2 Laboratory of Dynamical Meteorology, University P. and M. Curie, Paris, France UCL, December 2

Instabilities of jets Motivations, strategy and techniques Instabilities of jets in stratified rotating fluids Instabilities known/expected for jets in stratified rotating fluid: short-wave Kelvin-Helmholtz (KH) instability for strong vertical shears long-wave barotropic instability (BTI) long-wave baroclinic instability (BCI) symmetric inertial instability (II)

Instabilities of jets Motivations, strategy and techniques Interpretation of instabilities: Phase-locking and resonance of the linear waves (counter)propagating in the background flow instability. Possible types of propagating waves: Poincaré (P) or inertia-gravity Baroclinic and barotropic Rossby (R) Physical origin: "elasticity" of the iso-density surfaces (P), "elasticity" of the iso-pv surfaces (R). Remark: Equatorial waves - extra types: Kelvin and Yanai (mixed inertia-gravity) waves.

Instabilities of jets Motivations, strategy and techniques Wave origins of jet instabilities "Standard" instabilities KH: P-P resonance BTI: barotropic R - R resonance (standard: geostrophic) BCI: baroclinic R - R resonance (standard: geostrophic) "Non-standard" inertial instability (II) Trapped waves with negative eigen square frequency. essentially ageostrophic: needs Ro = O(); symmetric with resp. to along-jet translations; needs vertical variations.

Instabilities of jets Motivations, strategy and techniques Motivations Questions How along-jet modulations influence II? What is the relation of II to BCI and BTI? What is essentially ageostrophic BCI? How II saturates nonlinearly and what is the difference with the BCI/BTI saturation? II is endemic on the equator - differences between mid-latitude and equatorial II?

Instabilities of jets Motivations, strategy and techniques The strategy and techniques The strategy Models Detailed linear stability analysis in a wide range of Ro High resolution NS of the saturation initialized with the most unstable eigenmodes 2-layer Rotating Shallow Water, f - and equatorial β-plane; Primitive Equations The method stability analysis: pseudospectral collocation NS: finite-volume well-balanced for RSW, WRF for PE

2-layer RSW model on the f -plane Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane ( t + u x + v y )u fv + g x (h + h 2 ) =, ( t + u x + v y )v + fu + g y (h + h 2 ) =, t h + x (h u ) + y (h v ) =, ( t + u 2 x + v 2 y )u 2 fv 2 + g x (rh + h 2 ) =, ( t + u 2 x + v 2 y )v 2 + fu 2 + g y (rh + h 2 ) =, t h 2 + x (h 2 u 2 ) + y (h 2 v 2 ) =. (2.) u i, v i i =, 2 are x and y components of the velocity in layer i (layer on top of the layer 2), h i are the thicknesses of the layers, r = ρ ρ 2, f = const is the Coriolis parameter, and g is gravity.

Barotropic Bickley jet I Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane { h = H (x) = H h 2 = H 2 (x) = H 2 + δ tanh ( ) x L U (x) = U 2 (x) = ( V,2 (x) = V (x) = tanh 2 ( ) ) x L H, H 2 = const, L and δ - width and intensity of the jet, V = gδ fl - peak velocity. Dimensionless parameters: Bu = gh, Ro = gδ, d = H f 2 L 2 (fl) 2 2 H, r, H = H + H 2. gδ fl (2.2)

Barotropic Bickley jet II Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane H.. 8 6 4 2 2 4 6 8 v.. 8 6 4 2 2 4 6 8 PV 8 6 4 2 2 4 6 8 Background jet profile as a function of for δ L = 2 ; solid: layer ; dashed: layer 2.

Linearization of the symmetric problem Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane Small perturbations of the jet (2.2): h i = H i (x)+η i (x, t), u i = u i (x, t), v i = V (x)+v i (x, t), i =, 2, (2.) Solution is sought in the form (u i, v i, η i ) = (u i(x), v i (x), η i (x)) e iωt + c.c. a pair of coupled Schrödinger equations for the across-front velocities of the layers: ( ) ( ω 2 u + f (f + x V ) u 2 ) ( g x 2 H u + H 2 u 2 rh u + H 2 u 2 ) =. (2.4)

Eigenfrequencies Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane In terms of the barotropic and the baroclinic velocity components: ( ) ( ) ub u2 u = H u u +H 2 u 2. (2.) B H +H 2 (f + ω 2 x V )H b u b 2 x (H = f b u b ) 2 Hb u b 2 + g( r) Hb u b 2 Hb ub + g( r) 2 x (H u B ) Hb u b 2, (2.6) where H b = /H +/H 2. Relative vorticity of the jet x V sufficiently negative ω 2 < inertial (symmetric) instability.

A typical unstable mode for Bu =, Ro =, d = 2, r =. Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane Re(η )..2...2. Re(u ).2...2 Re(v ).7..2.2...2..2..7. Im(η ) Im(u ) Im(v ).2...2.2.2..

Stability diagram for small Ro Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane Re(w)/k.2... 2 4 6 k Im(w)..7..2 2 k 4 6 Left: phase speed Re(ω)/k as a function of k; Right: Growth rate Im(ω) as a function of k. Quasi-geostrophic jet: H =, Bu =, Ro =., d = 2, r =..

Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane Cross-section of the most unstable mode Re(η ) Im(η ).. 2. 2... 2. 2. Re(u ).2.2.4.6 2. 2. Im(u ).2.2.4.6 2. 2. Re(v ) Im(v ).. 2. 2... 2. 2. Solid (dashed): layer (2) barotropic instability

2D structure of the most unstable mode Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane 2. 2. 2. 2. 2. 2. 2. 2. Left(Right): upper(lower) layer barotropic instability.

Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane Cross-section of the second-unstable mode Re(η ).4.2.2 2. 2. Re(u ).4.2.2.4 2. 2. Re(v ).. 2. 2. Im(η ).4.2.2 2. 2. Im(u ).4.2.2.4 2. 2. Im(v ).. 2. 2. Solid (dashed): layer (2) baroclinic instability

2D structure of the third-unstable mode Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane 2. 2. 2. 2. 2. 2. 2. 2. Dipolar barotropic mode.

Stability diagram for large Ro Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane Re(w)/k 2.. 2 4 6 k Im(w).7..2 2 4 6 k Left: phase speed Re(ω)/k as a function of k; Right: Growth rate Im(ω) as a function of k. Strongly ageostrophic jet: H =, Bu =, Ro =, d = 2, r =.. BCI growth rate has non-zero limit at k.

Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane Cross-section of the most unstable mode Re(η )..2...2. Re(u )..2.. Re(v ).7..2.2 Im(η )..2...2. Im(u )..2.. Im(v ).7..2.2 Striking resemblance to D II mode.

2D structure of the most unstable mode Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane 7. 7. 2. 2. 2. 2. 7. 7. 2. 2. 2. 2. Essentially baroclinic, concentrated in the anticyclonic shear.

Introduction Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane Early stages of nonlinear evolution of AII: thickness

Introduction Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane Late stages of nonlinear evolution of AII: thickness

Loss of hyperbolicity Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane.. discrm. 2 2. 2 Time Time evolution of the minimum (over the whole calculation domain) of the discriminant of the characteristic matrix.

Energetics of the saturation Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane x Energy 2 Time Normalized deviation of the total (solid), kinetic (black dashed), and potential (gray dashed) energy from their initial values. "Inverse" (kinetic potential) budget.

Intensification of ageostrophic motions Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane 2 2 < div(v) > 2 Time Evolution of the norm of ageostrophic motions. Baroclinic (dotted) and barotropic (solid) component.

AII as a source of inertia-gravity waves Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane X/L c c 2 b B 2 2 2 c b2 c B2 2. 7. 2. 7. 2 22. 2 Time Hovmöller diagram of the baroclinic divergence field at y =. Typical values +. (black) and. (gray). Spectrum of y - averaged divergence norm at = ±2 integrated over supra-inertial motions: 6% of the whole integrated spectrum.

Introduction Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane Saturation of ageostrophic BTI: vertical shear and PV

Hyperbolicity loss Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane. discrm.. 2 Time Time evolution of the minimum (on the whole calculation domain) of the discriminant of the characteristic matrix.

Energetics of the saturation Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane.. Energy... 2 Time Normalized deviation of the total (solid), kinetic (black dashed), and potential (gray dashed) energy from their initial values. "Direct" (potential kinetic) budget.

Intensification of ageostrophic motions Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane 4 2 < div(v) > 8 6 4 2 2 Time Evolution of the norm of ageostrophic motions. Baroclinic (dotted) and barotropic (solid) component.

Saturation of the geostrophic BCI Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane.... T i.. 2 2..2.2.......2.2 Relative vorticity. Colors: lower, contours: upper layer.

Summary of the 2-layer f -plane results I Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane The leading long-wave instability of essentially ageostrophic jet is asymmetric inertial (AII). The most unstable mode is: baroclinic with dipolar cross-jet structure 2 localized at the anticyclonic side of the jet tends to the eigenmode of symmetric inertial instability at k, but has a larger growth rate At the nonlinear stage AII produces small-scale structures and strong shears leading to enhanced dissipation. Considerable emission of IGW observed.

Setup Stability analysis of the symmetric configuration Results of stability analysis in the 2D configuration Nonlinear saturation of asymmetric inertial instability (AII) Comparison with BTI and geostrophic BCI Summary: 2-layer Bickley jet on the f-plane Summary of the 2-layer f -plane results II The second-leading instability is barotropic (BTI) With diminishing Ro a mode swap takes place with barotropic instability becoming most unstable 2 monopolar baroclinic mode becoming second-unstable dipolar barotropic mode becoming third-unstable A region of wavenumbers exist where AII and BTI compete

Motivations and setup Results of the linear stability analysis Nonlinear saturation Motivations Inertial instability most natural in equatorial region due to vanishing f Zonal jets ubiquitous at equator, their instabilities determine much of dynamics Systematic studies of the instabilities of equatorial jets scarce Instabilities can be viewed as linear resonances of waves supported by the sytem - vast variety of equatorial waves, as compared to the f - plane

Motivations and setup Results of the linear stability analysis Nonlinear saturation Governing equations 2-layer RSW D it vi + βy e z v i + g (ρ i h + h 2 ) =, t h i + x (h i u i ) + y (h i v i ) =. (.) where the index i =, 2 refers to the upper and the lower layers, respectively, D it = t + u i x + v i y are advective derivatives in the layers, ρ, ρ 2 = ρ <, are the densities of the layers, and v i, h i are velocities and thicknesses of the layers.

Motivations and setup Results of the linear stability analysis Nonlinear saturation Background flow Barotropic Gaussian jet H (y) = H, H 2 (y) = H 2 He ()2, U (y) = 2U e ()2, U 2 (y) = 2U e ()2, V (y) = V 2 (y) =, where H H 2 and U = g H. The dimensionless βl 2 parameters are the Rossby number, the Burger number, and d i = H i H. We also introduce the parameters λ = H H (necessarily, d 2 λ), and d = H H 2. = Ro Bu (.2)

Motivations and setup Results of the linear stability analysis Nonlinear saturation H/H U/U. 6 6 2 6 6 H/H U/U. 6 6 2 6 6 ζ abs /f.8 6 6 ζ abs /f 2 4 6 6 Background flow with Bu =, ρ =. and d =.2 at Ro =. (left column) and Ro =. (right column) with λ = H H = Ro Bu = or., resp.. Top: thicknesses of the layers; Middle: zonal velocity profile (same in 2 layers); Bottom: profile of the absolute vorticity normalized by the planetary vorticity βy+ζ βy (same in 2 layers).

Motivations and setup Results of the linear stability analysis Nonlinear saturation Stable configuration Ro =.. Wave spectrum.

Motivations and setup Results of the linear stability analysis Nonlinear saturation Example of a phase-portrait: baroclinic and barotropic Yanai waves 6 6 6 9 6 6 9 6 9 6 6 9 6 6 6 6 9 6 6 9 9 6 6 9 Upper: - barotropic, Lower: - baroclinic

Motivations and setup Results of the linear stability analysis Nonlinear saturation Stability diagram for Ro =.27, ρ =., d =.2

Introduction Motivations and setup Results of the linear stability analysis Nonlinear saturation Evolution of stability diagrams from Ro =. to Ro =. and appearance of symmetric II.4.2 Im(w)/f Im(w)/f...2.....2 kl/2π.2......2 kl/2π.2......2 kl/2π.2...8.6.7..6 Im(w)/f Im(w)/f.4..2..4..2......2 kl/2π.2.. V. Zeitlin Asymmetric inertial instability

Motivations and setup Results of the linear stability analysis Nonlinear saturation Phase-portrait of the most unstable mode at Ro =. 6 6 6 6 6 Left: - upper, Right: - lower layer 6 6 6

Motivations and setup Results of the linear stability analysis Nonlinear saturation A résumé of the linear stability analysis in the two-layer model Barotropic Gaussian easterly jet in equatorial 2-layer model is stable at small Rossby numbers but loses stability at Ro.2. Dominant instabilities are due to phase-locking and resonances between a pair of barotropic, or a pair of baroclinic Yanai-type waves propagating on the background of the jet. Standard barotropic and baroclinic instabilities due to resonances of Rossby waves are present, but are weaker. With increasing Ro the baroclinic YY - instability overcomes the barotropic one, and the wavenumber of the most unstable mode diminishes. At high enough Ro this instability has nonzero growth rate at k =, giving a standard symmetric inertial instability. Yet the maximal growth rate is achieved at small but non-zero k.

Motivations and setup Results of the linear stability analysis Nonlinear saturation Saturation of the most unstable mode at Ro =.. Relative vorticity - upper layer. t = /βl t = 7/βL t = /βl 6 6 6 6 2 9 6 6 9 2 6 6 2 9 6 6 9 2 6 2 9 6 6 9 2 t = 4/βL t = 9/βL t = 28./βL 6 6 6 2 9 6 6 9 2 6 2 9 6 6 9 2 6 2 9 6 6 9 2

Motivations and setup Results of the linear stability analysis Nonlinear saturation Saturation of the most unstable mode at Ro =.. Relative vorticity - lower layer. t = /βl t = 7/βL t = /βl 6 6 6 6 2 9 6 6 9 2 6 6 2 9 6 6 9 2 6 2 9 6 6 9 2 t = 4/βL t = 9/βL t = 28./βL 6 6 6 2 9 6 6 9 2 6 2 9 6 6 9 2 6 2 9 6 6 9 2

Motivations and setup Results of the linear stability analysis Nonlinear saturation Saturation of the most unstable mode at Ro =.. Energy evolution. E/E..2 2 4 t βl Solid: - total, Dashed black: - kinetic, Dashed gray: - potential energy.

Motivations and setup Results of the linear stability analysis Nonlinear saturation Saturation of the most unstable mode at Ro =.. Vertical shear and dissipation t = /βl t = 7/βL t = /βl 6 6 6 6 2 9 6 6 9 2 6 6 2 9 6 6 9 2 6 2 9 6 6 9 2 t = 4/βL t = 9/βL t = 28./βL 6 6 6 2 9 6 6 9 2 6 2 9 6 6 9 2 6 2 9 6 6 9 2 Velocity shear u 2 u (thin black), enhanced dissipation zones (black) and zones of hyperbolicity loss (gray).

Motivations and setup Results of the linear stability analysis Nonlinear saturation Summary of the results at the equator The threshold of instability of easterly jets centered at the equator is at Ro /4, Dominant barotropic and baroclinic instabilities are due to (modified) Yanai waves, and not Rossby waves as on the mid-latitude tangent plane Classical symmetric inertial insability is a particular case of essentially ageostrophic baroclinic instability arising at high enough Ro Asymmetric inertial instability is stronger than symmetric one Nonlinear saturation of the dominant instability at large Ro produces much mixing and strong dissipation, for small Ro nonlinear oscillations of the jet observed.

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Goals and strategy Goals: To understand how the results of 2-layer model are modified for continuous stratification To understand how I I and B I interact in fully D simulations Strategy Detailed linear stability analysis of the barotropic Bickley jet by the collocation method in a wide range of parameters Direct numerical simulations of the saturation of the inertially unstable jet with WRF model

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet The model Continuously stratified primitive equations D t v h + f v h + h Φ tot ν z 2 v h =, D t θ tot κ z 2 θ tot =, z Φ tot g θ tot θ r =, h v h + z w =. (4.) Here ( v h, w ) - components of velocity, v h = (u, v), D t = t + v., f = f k, ν and κ - vertical (turbulent) viscosity and diffusivity (set to zero in linear stability analysis). Below: linear stratification θ (z) Φ (z). N 2 = g zθ θ r = const, θ r - a reference value.

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Background flow Barotropic Bickley jet: Balanced solution of (4.) Φ = Φ(y), u = ū(y), v = w = : Φ(y) = gh tan(), ū(y) = gh fl ( tan2 ()). (4.2)

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Scales and parameters L and gh - width and intensity of the jet, U = gh fl velocity. - peak Dimensionless parameters: Burger number Bu = ( ) N 2, f Rossby number Ro = U fl = gh, (fl) 2 Aspect ratio δ = H L, Ekman number γ = ν fh 2. Necessary condition for inertial instability: f q < (at the anticyclonic side of the jet) if Ro > 4, where q = f yū is the vertical component of the absolute vorticity of the jet.

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Barotropic Bickley jet. Φ/Φ U/U.. 2. 2. 2. 2. 2 ζ/f 2. 2. Cross-section of a jet with Ro = 2, Bu =.9 4, δ =.8 4. Top left: geopotential, Top right: velocity, Bottom: Vertical component of the normalized absolute vorticity.

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Linear stability analysis Workflow Linearize equations (4.) about the background jet Look for a solution in the form e i(ωt kx mz) For each vertical mode m, and each downstream wavenumber k treat the eigenvalue problem for the phase speed c = ω/k by collocation method in y Plot stability diagrams for the phase speed Re(c) and growth rate Im(ω) as functions of k for each m

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Stability diagram for small Ro..7.7 Im(w)/f. Re(c)/Uo..2.2....2 kl/2π....2 kl/2π Linearly unstable modes of the jet at Ro =., Bu =.9 4, δ =.27 4 and γ =. Left: Normalized growth rate and Right: Normalized phase speed as functions of the downstream wavenumber k. Black: Vertical wavenumber m = ( 2π L ). Gray: Vertical wavenumber m = 2( 2π L ).

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet 2D structure of the most unstable modes 4 2 2 4.7..2.2..7 x 4 2 2 4.7..2.2..7 x 2D structure of the two most unstable modes of the jet with Ro =., Bu =.9 4, δ =.27 4 and γ =. Vertical wavenumber m = ( 2π L ). Isolines of geopotential and velocity (arrows) anomalies. Positive (negative) anomalies: gray (black). L x = π k. Left: most unstable mode (k =.44( 2π L )). Right: less unstable mode (k =.8( 2π L )).

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Stability diagram for large Ro. Im(w)/f.4..2. Re(c)/Uo.7..2....2 kl/2π....2 kl/2π Linearly unstable modes of the jet at Ro = 2, Bu =.9 4, δ =.8 4 and γ =. Left: Normalized growth rate and Right: Normalized phase speed as functions of the downstream wavenumber k. Light gray dots: Vertical wavenumber m = ( 2π L ). Gray dots: m = ( 2π L ). Black dots: m = 7( 2π L ). Circles: m = ( 2π L ).

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet 2D structure of the most unstable modes m = ( 2π L ) m = ( 2π L ) 4 2 2 4 4 2 2 4.7..2.2..7 x.7..2.2..7 x m = 7( 2π L ) m = ( 2π L ) 4 2 2 4 4 2 2 4.7..2.2..7 x.7..2.2..7 x Most unstable modes of the jet with Ro = 2, Bu =.9 4, δ =.8 4 and γ =.

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Constructing initial state To avoid influence of vertical boundaries: barotropic unstable jet in the center of the domain, becoming stable at top/bottom: U(y, z) = U ( tan 2 ())e ( z H tot /2 H tot / )2, U fl = 2 (U = m/s). Thermal wind balance horizontal gradients of temperature in the regions of vertical shear. L = km, f.7 4 s. Computational domain: l y = L (walls - negligible influence), l x = L. Top: H tot = 9km fl/n, N =.s, Bu =.9 4. Wind/temperature from thermal wind balance. Perturbation: white noise in cross-stream velocity. Resolution: dx, dy =.L, dz =.6H tot, f dt =..

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Background jet.7.7 z/lz. z/lz..2.2 2. 2. 2. 2.... 2 2. Initial jet for the WRF simulation. Ro = 2. Bu =.9 4. Left: Temperature θ (thin black lines) and downstream velocity U (black lines). Contour interval 2K for θ and 2m/s for U. Right: Vertical vorticity ζ = x v y u. Positive: black. Negative: gray. Contour interval.2 f, starting from.2 f.

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Evolution of the along-jet Fourier modes 4 Log[a j /a j ()] 2 2 4 6 7 Time(days) Evolution of the normalized kinetic energy of zero < a > (black line), first < a >(light gray line) and second < a 2 >(gray line) modes of the along-jet Fourier transform. < a (t = ) > < a (t = ) > < a 2 (t = ) >.

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Evolution of vorticity: across-flow view day at hms day at hms day at hms.7.7.7 z/lz. z/lz. z/lz..2.2.2 2. 2.... 2 2. 2. 2.... 2 2. 2. 2.... 2 2. day 7 at hms day 9 at hms day at hms.7.7.7 z/lz. z/lz. z/lz..2.2.2 2. 2.... 2 2. 2. 2.... 2 2. 2. 2.... 2 2.

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Evolution of vorticity: along-flow view day at hms day at hms day at hms 4 2 2 4 4 2 2 4 4 2 2 4 pi pi/2 pi/2 pi x/λ pi pi/2 pi/2 pi x/λ pi pi/2 pi/2 pi x/λ day 7 at hms day 9 at hms day at hms 4 2 2 4 4 2 2 4 4 2 2 4 pi pi/2 pi/2 pi x/λ pi pi/2 pi/2 pi x/λ pi pi/2 pi/2 pi x/λ

The leading instability of quasi-geostrophic jet is standard baroclinic (B I) The most unstable B I mode is cross-flow monopolar The leading long-wave instability of essentially ageostrophic jet is inertial (I I). A transition takes place at Ro.2. I I coexists with standard B I then. In a wide range of (high) vertical wave numbers the asymmetric I I dominates At very high vertical wave numbers symmetric I I is the strongest. The most unstable I I mode is cross-flow dipolar, concentrated at the anticyclonic V. Zeitlin Asymmetric sideinertial of the instability jet Introduction Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Summary of continuously stratified case, linear stability

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Summary of continuously stratified case, nonlinear saturation I I rapidly develops from a small-amplitude white-noise perturbation Both symmetric and asymmetric I I patterns develop, Nonlinear saturation of I I produces homogenization of the cross-stream gradient of the vorticity in the unstable region At later stages B I takes over. Its saturation leads to formation of coherent vortices downstream. Persistent internal gravity wave emission accompanies the saturation process Vortex street formation results from the development of B I of the jet modified by the saturation of the inertial instability

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet Overall conclusions: AII dominant in a large range of vertical wavenumbers Saturates, like symmetric II, by massively producing overturning motions and enhanced dissipation and mixing Very high vertical wavenumbers needed to achieve the dominance of symmetric II II evolves simultaneously with /gives in to conventional barotropic instability Remarks: Changing vertical resolution in essentially ageostrophic regimes may qualitatively change the output Similar program for centrifugal instability - similar results (Lahaye & Zeitlin, 2).

Goals, strategy and setup Results of the linear stability analysis Nonlinear evolution of the unstable jet References B. Ribstein, R. Plougonven and V. Zeitlin "Inertial, baroclinic and barotropic instabilities of the Bickley jet in two - layer rotating shallow water model", Phys. Fluids, 2 F. Bouchut, B. Ribstein and V. Zeitlin "Inertial vs baroclinic instability of the Bickley jet in continuously stratified rotating fluid" J. Fluid Mech, 2 - under consideration B. Ribstein, A.-S. Tissier and V. Zeitlin "Barotropic, baroclinic, and inertial instabilities of the Gaussian jet on the equatorial β-plane in rotating shallow water model" Phys. Fluids, 2 - under consideration