and their nonlinear evolution in the two-laer rotating shallow water model. Influence of the lower laer and of the topograph. B. Ribstein and V. Zeitlin Laborator of Dnamical Meteorolog, Univ. P.et M. Curie, Paris, France Sub-mesoscale Ocean Processes, Toronto 21
Plan 1 Introduction Introduction 2 The model Scalings and background flow Boundar conditions Numerical settings Epectations Selected results 4 Motivation and general setting of the DNS Saturation of competing FF and RF instabilities Saturation of instabilities in the presence of topograph 5
Densit fronts reminder Densit fronts: ubiquitous in nature and eas to reproduce in the lab characteristicall unstable following classics (Griffiths, Killworth & Stern, 1982), the instabilities of DF are traditionnall studied in the framework of 1- or 2-laer rotating shallow water (RSW) models ; result from phase-locking and resonance of characteristic frontal waves recent progress: detailed numerical linear stabilit analsis and high-resolution DNS of nonlinear saturation (Gula, Zeitlin & Bouchut, 21).
Motivation Not well- understood: the role of the bottom laer the role of topograph details nonlinear saturation Main motivation: To investigate how the classical instabilit of the double densit fronts, resulting from a resonance between two frontal waves propagating along the respective fronts, interacts with other long-wave instabilities appearing due to the active lower laer and topograph and, respectivel, Rossb and topographic waves which are activated in the sstem.
Goals Introduction We want: to give a complete classification of the instabilities of double densit fronts in the presence of an active lower laer and shelf-like topograph to intercompare them and to identif the dominant one and possible instabilit swaps in the parameter space to identif and intercompare different saturation patterns Program realized in:, 21, J. Fluid Mech., 716, 528-565.
Methodolog and tools We follow previous work (Scherer & Zeitlin 28; Gula & Zeitlin 21; Gula, Zeitlin & Bouchut 21) and add combined effects of baroclinicit and bottom topograph: Densit fronts: 2-laer RSW with outcropping interface. Topograph: escarpment beneath the upper-laer current. Steep topograph: horizontal scale width of the current. Straight fronts with velocit in geostrophic balance: eact solutions. Linear stabilit: collocation method. Unstable modes: resonances between eigenmodes. Unstable modes initialization of numerical simulations with new-generation well-balanced high resolution finite-volume scheme (Bouchut & Zeitlin 21).
The model Scalings and background flow Coupled densit fronts with nontrivial bathmetr. ρ u 2LR d h 1 1 1. H f 2^z g ρ 2 u = 2 h 2 rh z= b 2aR d αoh R d : deformation radius, L and a: nondimensional widths of the balanced current and of the escarpment. r: depth ratio, α : non-dimensional amplitude of the bathmetr.
Equations of the model The model Scalings and background flow ( t + u i + v i )u i fv i + Π i =, ( t + u i + v i )v i + fu i + Π i =, t h i + ((h i b δ i2 )u i ) + ((h i b δ i2 v i ) =. u i, v i (i = 1, 2) - - and - components of the velocit in the laers (laer 1 on top of the laer 2); h 1, h 2 b - thicknesses of the laers, δ ij -Kronecker delta; ρ = ρ 1 ρ 2 1 - densit ratio, f = const - Coriolis parameter, g - gravit. Geopotentials of the laers (1, 2): (1) Π 1 = g(h 1 + h 2 ), Π 2 = g(ρh 1 + h 2 ). (2)
The model Scalings and background flow Intrinsic scales: Length: radius of deformation R d = gh(1 ρ)/f, Time: 1/f. dimensionless wavenumber ɛ = 2πR d /λ Scalings: cross-stream coordinate R d, downstream coordinate λ/2π = R d /ɛ time t 1/ɛf. width of the current: 2R d L, L = O(1). bathmetr variations: R d a cross-stream velocities ɛ gh(1 ρ), and downstream velocities gh(1 ρ) Ro = 1 2L.
The model Scalings and background flow Non-dimensional equations of the model ( t + u i + v i )u i v i + Π i =, ɛ 2 ( t + u i + v i )v i + u i + Π i =, t h i + ((h i α r b δ i2 )u i ) + ((h i α r b δ i2 )v i ) =, () Π 1 = h 1+rh 2 1 ρ, Π 2 = ρh 1+rh 2 1 ρ.
Backround flow Introduction The model Scalings and background flow Background flow (ū i, v i, h i ) is a geostrophicall balanced, parallel to the ais densit current terminating at ±L, with no mean flow in the lower laer: ū 1 = ū = h, Π 1 = h, Π 2 =, ū 2 = v 2 = v 1 =. (4) 1 = h is the background thickness of the upper laer, h(±l) =, otherwise h() is arbitrar There is no variation of bathmetr beond the outcroppings a < L: b = 1, < a, b =, > a. (5)
Constant-PV currents The model Scalings and background flow If potential vorticit is constant Q in the upper laer, h Q h + 1 =, and ( 1 Q < 1 : h = 1 cosh( Q) Q cosh(l ( Q) Q < : h = 1 Q 1 cos( Q ) cos(l Q ) ) ) (, L = 1 1 + Q(2 Q ln Q 1 Q, L = 1 Q cos ( 1 1 + Q Q = : h = 1 (/L) 2, L = 2 (6) For Q =.5 - a configuration to be used below for illustrations, L 1.86 and Ro.27. )
The model Scalings and background flow Cross-section of the background flow h i u i 2 1.5.5.5 1 /L 1.5.5.5.5 1 /L Background flow for constant PV Q =.5 in the upper laer. Densit ratio ρ =.5. Depth ratio r = 1. Topograph: α = 5 and a =.5L. Upper panel: interface (dashed), free surface (solid) and topograph (thick). Lower panel: downstream velocities of the laers 2 (dashed) and 1 (solid).
Boundar conditions Boundar conditions Numerical settings Epectations Selected results Linear stabilit: small perturbation (u i, v i, h i ). Boundar conditions: h + h 1 = and dl ± = v 1 at = ±L + λ ±, (7) dt ±L - locations of the free streamlines of the balanced flow, λ ± (, t) - perturbations of the free streamlines. Another boundar condition: for the lower laer, the continuit of the solution at ±L. Beond the outcropping: eponential deca of the pressure perturbation on both sides of the double front boundar conditions at the outcroppings entire linear eigenproblem solel at the interval [ L, L].
Boundar conditions Numerical settings Epectations Selected results Comment on boundar conditions Spectrum of the linearized problem in the cross-flow direction: discrete + continuous. First: free inertia-gravit waves. Second: trapped waves eponentiall decaing out of the densit fronts. B imposing the deca boundar condition we filter out free inertia-gravit waves and concentrate uniquel on the trapped modes - consistent with our interest in long-wave instabilities. Onl the instabilities resulting from the resonances between the trapped modes will be captured, radiative instabilities due to the resonances with free inertia-gravit waves are ecluded. A priori justification: condition of efficient emission of inertia-gravit waves b a PV anomal is Ro 1, while we work with Ro < 1. A posteriori justification: no radiative instabilities observed in DNS.
Boundar conditions Numerical settings Epectations Selected results Numerical linear stabilit analsis Method: pseudospectral collocation (Trefethen 2) matri eigenproblem for the phase speed c of the perturbation, MATLAB routine "eig", Boundar conditions: continuit of all variables at = ±L, continuit of the lower-laer pressure at = ±L and eponential deca out of the front, Discretization: Chebshev collocation points { i = L cos(jπ/n), j =, 1,.., N}. Numerical convergence tpicall for N=5, sstematic checks with double resolution. Chebshev differentiation matri for discrete differentiation, Topograph: escarpment with a linear slope, Treatment of spurious soloutions (singular modes): filtering based on slope limiters + increase of resolution.
Wave species of the flow Boundar conditions Numerical settings Epectations Selected results Flow with constant Q in the upper laer and bottom escarpment: Poincaré (inertia-gravit) modes in both laers, Rossb modes in the lower laer (no PV gradients in the upper laer), Frontal modes, trapped in the vicinit of the free streamlines in the upper laer, Topographic waves in the lower laer, trapped b the varing bathmetr. Instabilities: resonances between the eigenmodes of the linearized problem. Resonances crossings of dispersion curves (Cairns 1979).
Epectations Introduction Boundar conditions Numerical settings Epectations Selected results Epect following resonances and related instabilities : 1 the barotropic resonances of the upper-laer modes between : two frontal waves (FF); a Poincaré and a frontal wave (P1F); two Poincaré waves (P1P1). 2 the baroclinic resonances of the modes of different laers between: a frontal upper wave and a lower Rossb wave (RF); a frontal upper wave and a lower topographic wave (TF); an upper Poincaré wave and a lower Rossb wave (P1R); an upper Poincaré wave and a lower topographic wave (P1T); a frontal upper wave and a lower Poincaré wave (P2F); upper and lower Poincaré waves (P1P2).
Boundar conditions Numerical settings Epectations Selected results Stabilit diagram: ver deep lower laer with Q =.5.6 ε Im(c) FF.4.2 TF PF RP,TP RP,TP RP,TP 1 2 4 5 6 7 8 9 1 ε.6 ε Im(c).4.2 FF TF TF.25.5.75 1 1.25 ε Densit ratio ρ =.5. Depth ratio r = 1. Topograph: a =.5L, α = 5. Gra: unstable. Black: stable. Waves: I - inertial; F - frontal; R - Rossb, T - topographic. Bottom: zoom.
Boundar conditions Numerical settings Epectations Selected results Most unstable mode: FF resonance 1 1 /L /L pi pi/2 pi/2 pi pi pi/2 pi/2 pi 2D structure of the most unstable mode (ɛ =.59): resonance between two frontal waves in the upper laer. Left: Isobars (contour interval.5) and velocit field of the perturbation in the upper laer. Right: Isobars (contour interval.1, starting from ±.1) and velocit field of the perturbation in the lower laer. Positive (negative) pressure anomalies: black (gra) lines. v 2 ma. v 1 ma.
Boundar conditions Numerical settings Epectations Selected results Stabilit diagram: moderatel deep lower laer.6 ε Im(c).4 FF.2 RF TF RF RP,TP TF PF RP,TP RP,TP 1 2 4 5 6 7 8 9 1 ε.6 ε Im(c).4.2 FF RF TF RF TF.25.5.75 1 1.25 ε Same background flow with the depth ratio r = 1 and the topograph parameter α = 5. Bottom: zoom in the stabilit diagram.
Boundar conditions Numerical settings Epectations Selected results Second unstable mode: RF resonance 1 1 /L /L pi pi/2 pi/2 pi pi pi/2 pi/2 pi 2D structure of the unstable mode with ɛ =.77. Resonance between a frontal wave (upper laer) and a Rossb wave (lower laer). Perturbation with Re(c) <. v 2 ma.5 v 1 ma.
Boundar conditions Numerical settings Epectations Selected results Net unstable mode: TF resonance 1 1 /L /L pi pi/2 pi/2 pi pi pi/2 pi/2 pi 2D structure of the unstable mode with ɛ = 1.15. Resonance between a frontal wave (upper laer) and the first topographic mode (lower laer). Isobars of the perturbation in the lower laer (contour interval.5, starting from ±.5). v 2 ma.5 v 1 ma.
Boundar conditions Numerical settings Epectations Selected results Unstable mode corresponding to TF resonance with second topographic mode 1 1 /L /L pi pi/2 pi/2 pi pi pi/2 pi/2 pi 2D structure of the unstable mode with ɛ =.842. Resonance between a frontal wave (upper laer) and the second topographic mode (lower laer). v 2 ma.5 v 1 ma.
Boundar conditions Numerical settings Epectations Selected results Unstable mode corresponding to P1R resonance /L pi pi/2 pi/2 pi /L pi pi/2 pi/2 pi 2D structure of the unstable mode with ɛ =.577. Resonance between a Poincaré mode in the upper laer and a Rossb wave in the lower laer. Perturbation Re(c) <. Isobars of the perturbation in the lower laer (contour interval.5, starting from ±.5). v 2 ma.55 v 1 ma.
Boundar conditions Numerical settings Epectations Selected results Stabilit diagram: shallow lower laer ε Im(c).7.6.5.4. PF.2 PF RF.1 RP,TP PP FF RF,TF RP,TP RP,TP RFPF 1 2 4 5 6 7 8 9 1 ε Stabilit diagram for the same background flow with the depth ratio r = 2 and the topograph parameter α = 1.4.
Boundar conditions Numerical settings Epectations Selected results Unstable mode corresponding to P2F resonance /L pi pi/2 pi/2 pi /L pi pi/2 pi/2 pi 2D structure of the unstable mode with ɛ =.885. Resonance between a Poincaré mode in the lower laer and a frontal wave in the upper laer. Perturbation with Re(c) <. Isobars of the perturbation in the lower laer (contour interval.1, starting from ±.1). v 2 ma.27 v 1 ma.
Boundar conditions Numerical settings Epectations Selected results Eample of competing instabilities.6 FF RF ε Im(c).4 RP.2 RP RP PF 1 2 4 5 6 7 8 9 1 ε.6 ε Im(c).4.2 FF RF.25.5.75 1 1.25 ε Stabilit diagram for the flat-bottom background flow with Q =.6, ρ =.5, and depth ratio r = 2. Bottom: zoom in the stabilit diagram.
Motivations/questions Motivation and general setting of the DNS Saturation of competing FF and RF instabilities Saturation of instabilities in the presence of topograph how the active lower laer influences the nonlinear evolution of the coupled densit fronts established in the framework of equivalent 1-laer model (Scherer & Zeitlin 28)? what are the differences in saturation of (FF) instabilit and of its rival, the (RF ) instabilit? how the presence of the second densit front/absence of the boundar (coast) influences the saturation of the (RF) instabilit observed in the case of the coastal current with similar settings (Gula, Zeitlin & Bouchut21)? how the presence of escarpment beneath the fronts changes the scenarii of saturation?
Numerical settings Motivation and general setting of the DNS Saturation of competing FF and RF instabilities Saturation of instabilities in the presence of topograph Numerical scheme: Finite-volume, well balanced, shock-capturing for 2-laer RSW with free upper surface (Bouchut & Zeitlin 21). Initialization/resolution: Initialization: basic flow + perturbation of the amplitude 1% of the ma thickness of the unperturbed upper laer. Perturbation: an unstable mode. Boundar conditions: sponges cross-stream, periodicit downstream. Resolution: tpicall.67 R d, control simulations with double resolution
Caveats of 2-laer model: Motivation and general setting of the DNS Saturation of competing FF and RF instabilities Saturation of instabilities in the presence of topograph Strong vertical shears loss of hperbolicit (phsicall: KH instabilities). Numerical scheme copes well with them: strong gradients trigger enhanced numerical dissipation and the scheme cures itself, the non-hperbolic zones remaining localized and eventuall disappearing. Rankine-Hugoniot conditions for the model are not complete, etra ad hoc hpotheses are needed to determine weak solutions (shocks). Our scheme: laerwise momentumn conservation
Motivation and general setting of the DNS Saturation of competing FF and RF instabilities Saturation of instabilities in the presence of topograph Benchmark: comparison with the results of the linear stabilit analsis σ t 7 6 5 4 2 1 1 2 4 5 6 7 log( v ) 1 Comparison of the growth rate of the FF instabilit at the initial stages of the direct numerical simulation initialized with the most unstable mode with the predictions of the linear stabilit analsis: logarithm of the norm of the cross-stream velocit in the upper laer vs time normalized b the linear growth rate.
Introduction Nonlinear evolution of the FF instabilit Motivation and general setting of the DNS Saturation of competing FF and RF instabilities Saturation of instabilities in the presence of topograph t=/f t=84/f t=1/f t=16/f 6 6 6 6 6 6 6 6 t=2/f t=258/f t=292/f t=8/f 6 6 6 6 6 6 6 6 Thickness h 1 of the upper laer (black). Contour interval.2, starting at.2. Pressure Π 2 (gra) of the lower laer. Contour interval.5, starting at rh ±.5 (+/- anomal: solid/dashed).
Motivation and general setting of the DNS Saturation of competing FF and RF instabilities Saturation of instabilities in the presence of topograph Energ balance of saturating FF instabilit 4 1 1 1.5 2 E/E E/E.5 2 5 1 15 2 25 5 4 Time 4 5 1 15 2 25 5 4 Time Left: Normalized deviation of the total (solid), kinetic (black dashed) and potential (gra dashed) energ from initial values. Right: Normalized deviations of the kinetic ρ i h i /2v 2 i (solid) and potential ρ i ghi 2 /2 (dashed) of laers 1 (black) and 2 (gra). Echange ρ 1 gh 1 h 2 (solid dotted) and total (dark gra) energies.
Introduction Nonlinear evolution of the RF instabilit Motivation and general setting of the DNS Saturation of competing FF and RF instabilities Saturation of instabilities in the presence of topograph t=/f t=8/f t=11/f t=16/f 6 6 6 6 6 6 6 6 t=18/f t=21/f t=242/f t=2/f 6 6 6 6 6 6 6 6 Thickness h 1 of the upper laer (black). Contour interval.2, starting at ±.2. Pressure Π 2 (gra) of the lower laer. Contour interval.5, starting at rh ±.5 (+/- anomal: solid/dashed).
Motivation and general setting of the DNS Saturation of competing FF and RF instabilities Saturation of instabilities in the presence of topograph Energ balance of saturating RF instabilit 1 1 4 1 E/E.5.5 E/E 2 2 5 1 15 2 25 5 4 Time 4 5 1 15 2 25 5 4 Time Left: Normalized deviation of the total energies from their initial values. Right: Evolution of different energ components.
Introduction Motivation and general setting of the DNS Saturation of competing FF and RF instabilities Saturation of instabilities in the presence of topograph Saturation of FF instabilit over escarpment t=/f t=1/f t=15/f t=2/f t=25/f t=/f t=4/f t=5/f Thickness h 1 of the upper laer (black). Contour interval.2, starting at ±.2. Pressure Π 2 (gra) of the lower laer. Contour interval.15, starting at rh ±.15 (+/-: solid/dashed).
Motivation and general setting of the DNS Saturation of competing FF and RF instabilities Saturation of instabilities in the presence of topograph Saturation of RF instabilit with topograph t=25/f t=/f t=5/f t=4/f Nonlinear evolution of the RF instabilit over escarpment. Thickness h 1 of the upper laer (black). Contour interval.2, starting from.2. Pressure Π 2 (gra) of the lower laer. Contour interval.1, starting at rh ±.1 (positive/negative pressure anomal: solid/dashed).
Introduction Motivation and general setting of the DNS Saturation of competing FF and RF instabilities Saturation of instabilities in the presence of topograph Nonlinear development of the TF instabilit t=/f t=2/f t=25/f t=/f t=5/f t=4/f t=45/f t=5/f Thickness h 1 of the upper laer (black). Contour interval.2, starting at.2. Pressure Π 2 (gra) of the lower laer. Contour interval.5, starting at rh = ±.5 (+/- : solid/dashed).
Energ balance of TF instabilit Motivation and general setting of the DNS Saturation of competing FF and RF instabilities Saturation of instabilities in the presence of topograph 1 1 5 E/E 7.5 1 6 5 2.5 2.5 5 E/E.5.5 7.5 5 1 15 2 25 5 4 45 5 Time 5 1 15 2 25 5 4 45 5 Time Energ balance of the developing TF instabilit: etremel small dissipation.
Linear stabilit analsis: Leading long-wave barotropic FF instabilit dominant for deep lower laers, ma be overcome b the baroclinic RF instabilit when the depth of the lower laer decreases, including asmmetric decrease in depth due to topograph. Topograph renders the FF instabilit propagative. Specific long-wave topographic TF instabilit arises. In the configuration with centered escarpment it is never dominant. For shallow (partiall shallow due to topograph) lower laers short-wave Kelvin-Helmholtz tpe instabilities become dominant.
Nonlinear saturation of long-wave instabilities: FF instabilit over the flat bottom saturates b reorganizing the flow in a sstem of co-rotating ellipsoidal quasi-barotropic vortices, the rodons. FF instabilit over escarpment saturates b reorganizing the flow into two rows of quasi-circular quasi-barotropic vortices on both sides of the escarpment. RF instabilit saturates b forming transient upper-laer monopolar and lower-laer dipolar vortices which, after a stage of a sideward motion reorganize themselves in a secondar mean current, shifted with respect to the initial one and eperiencing subsequent secondar instabilities. TF instabilit saturates forming a stead finite-amplitude nonlinear hbrid fronto-topographic wave