understanding large-scale atmospheric and oceanic flows ... · v. zeitlin, non-homogeneous fluids...

Lecture 3:Applications to the

ocean,density-driven

coastal currents

Motivation

Approach

Passive lower layerPreliminaries

Linear stability

Nonlinear saturation

Manifestation of theinstability in initial-valueproblem

Active lower layerPreliminaries

Linear stability


Role of the vertical shear(KH) instability

Summary andconclusions

Literature

Understanding large-scale atmosphericand oceanic flows with layered rotating

shallow water models

V. Zeitlin,

Non-homogeneous Fluids and Flows, Prague, August2012



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

PlanMotivation

Approach

Passive lower layerPreliminariesLinear stabilityNonlinear saturationManifestation of the instability in initial-value problem

Active lower layerPreliminariesLinear stabilityNonlinear saturationRole of the vertical shear (KH) instability

Summary and conclusions

Literature



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Ingredients

I Multi-layer shallow-water models;I Exhaustive linear stability analysis by the collocation

method;I High-resolution numerical simulations of nonlinear

evolution with new-generation finite-volume code.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Workflow

I Choose a model: 1.5 or 2 -layer (or more!);I Choose bathymetry;I Choose balanced profiles of velocity/interface;I Analyse linear stability: unstable modes, growth

rates;I Initialise nonlinear simulations with the unstable

modes, study saturation;I Look how instabilities manifest themselves in

initial-value problem.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Typical configuration

y = 0

y = −L

ρ1

ρ2

f2

y

H1(y) U1(y)

H2(y)U2(y)



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

RSW equations with coast (no bathymetry)

Equations of motion:

ut + uux + vuy − fv + ghx = 0,vt + uvx + vvy + fu + ghy = 0,

ht + (hu)x + (hv)y = 0. (1)

Boundary conditions:

H(y) + h(x , y , t) = 0, DtY0 = v at y = Y0 , (2)

where Y0(x , t) is the position of the free streamline, Dt isLagrangian derivative.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Balanced flows:u = U(y), v = 0, and h = H(y),

U(y) = −gf

Hy (y) (3)

exact stationary solution.

−1 00

0.25

0.5

0.75

H1(y)

y−1 0

−0.5

0

0.5

U1(y)

y

Figure: Examples of the basic state heights (left) and velocities(right) for constant PV flows with U0 = −sinh(−1)/cosh(−1)(thick line), U0 = 1/2 (dotted) and a zero PV flow (dash-dotted)



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Non-dimensional linearized system:

ut + Uux + vUy − v = − hx ,vt + Uvx + u = − hy ,

ht + Uhx = −(Hux + (Hv)y ).(4)

Linearized boundary conditions:

I

Y0 = − hHy

∣∣∣∣y=0

, (5)

I continuity equation evaluated at y = 0.

The only constraint is regularity of solutions at y = 0.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

PV of the mean flow

Q(y) =1− UyH(y)

, (6)

Geostrophic equilibrium⇒

Hyy (y)−Q(y)H(y) + 1 = 0, with{

H(0) = 0Hy (0) = −U0,

(7)

U(0) = U0 is the mean-flow velocity at the front.



coastal currents

Motivation

Approach


Linear stability




Linear stability




LiteratureWave Number

U0

0 1 2 3 4 5 6 7 8 9 10

0.5

0.55

0.6

0.65

0.7

0.75

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Figure: Stability diagram in the (U0fL , k) plane for the constantPV current. Values of the growth rates in the right column.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Dispersion diagram: stable flow

0 1 2 3 4 5 6 7 8 9 10

0

1

c

k

K

F

Pn

Pn

Figure: Dispersion diagram for U0 = −sinh(−1)/cosh(−1) andQ0 = 1.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Dispersion diagram: unstable flow

0

0.5

1

c K

F

Pn

Pn

2 4 6 8 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

k

σ

Figure: Dispersion diagram for U0 = 0.5 and Q0 = 1.Crossings of the dispersion curves in the upper panelcorrespond to instability zones in the lower panel.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

The most unstable mode: Kelvin-Frontalresonance

y

x−1

0

Figure: Height and velocity fields of the most unstable modek = 3.5.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Saturation of the primary instabilityy

x

t= 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

y

x

t= 33

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Figure: Height and velocity fields of the perturbation at t = 0(left) and t = 30 (right). Kelvin front is clearly seen at thebottom of the right panel.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Kelvin wave breaking

1 2 3 4 5 6 7−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1t= 22.5026

u

x

Figure: Evolution of the tangent velocity at y = −L (at the wall)for t = 0,2.5,5,7.5,10,12.5,15,17.5,20,22.5 (from lower toupper curves)



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Secondary instabilityy

x0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

−0.8

−0.6

−0.4

−0.2

0

0.2

y

x0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Figure: Height and velocity fields of the secondary perturbationat t = 335, t = 500 (right).



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Reorganization of the mean flow

−1 −0.5 0 0.50

0.05

0.1

0.15

0.2

0.25

0.3

Hzonal

y

−1 −0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

y

uzonal

Figure: Evolution of the mean zonal height (left) and meanzonal velocity (right): Initial state t = 0 (dashed line), primaryunstable mode saturated at t = 40 (dash-dotted line), latestage t = 300 (thick line).



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Stability diagram of the reorganized flow

Figure: Dispersion diagram of the eigenmodes correspondingto the basic state profile of the flow at t = 335, at the beginningof the secondary instability stage (see Fig. 9).



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Most unstable mode of the reorganized flow

y

x−1

0

Figure: Height and velocity fields of the most unstable mode offigure 10 for k = k0. Only one wavelenght is plotted. Note thesimilarity with the mode observed in the simulation, Fig. 8



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Instability in Cauchy problem

c

y

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−1

−0.5

0

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

c

y

−0.1 0 0.1 0.2 0.3−1

−0.5

0

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

Figure: y − c diagram of the height field at t = 45 of thedevelopment of initially localised perturbation (dotted) forlinearly stable (upper) and unstable (lower) current.Phase-locking of frontal and Kelvin waves in the lower panel



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Equations of motion

Djuj − fvj = − 1ρj ∂xπj ,Djvj + fuj = − 1ρj ∂yπj ,

Djhj +∇ · (hjvj) = 0,(8)

j = 1,2: upper/lower layer, (x , y), hj(x , y , t) - depths ofthe layers, πj , ρj - pressures, densities of the layers,

∇πj = ρjg∇(sj−1h1 + h2), s = ρ1/ρ2. (9)



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Stationary solutionsBalanced flow with depths Hj(y) and velocities Uj(y):

∂yHj = (−1)j−1 fg′ (U2 − sj−1U1), (10)

Linearization/nondimensionalization:

∂tuj + Uj∂xuj + vj∂yUj − vj = −∂x(sj−1h1 + h2),∂tvj + Uj∂xvj + uj = −∂y (sj−1h1 + h2),

∂thj + Uj∂xhj + Hj∂xuj = −∂y (Hjvj).(11)



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Boundary conditions

I Upper layer: same as in 1.5-layer case,I Lower layer: for harmonic perturbations

(uj(x , y), vj(x , y),hj(x , y)) = (ũj(y), ṽj(y), h̃j(y)) ei(kx−ωt),(12)

Filtering of outer inertia-gravity waves, decaycondition:

∂y (sh1 + h2) = −k(sh1 + h2) at y = 0.

Key parameters:U0, the non-dimensional velocity of the upper layer at thefront location y = 0, equivalent to Rossby number, aspectratio r = H1(−1)/H2(−1), and stratification s = ρ1/ρ2.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Configurations considered:

I Bottom layer: initially at rest (U2 = 0),I Upper layer: with constant PV.

Two classes of flows: barotropically stable/unstable, i.e.stable/unstable in the 1.5 - layer limit.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Barotropically stable case

Figure: Dispersion diagrams for s = .5. (a) r = 10, (b) r = 2,(c) r = 0.5. Horizontal scale of the bottom panel shrinked toshow short-wave KH instabilities.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Barotropically unstable case

Figure: Dispersion diagrams for s = 0.5 and for Rd = 1. (a)r = 10, (b) r = 5, (c) r = 2 . The horizontal scale of the panelsshrinked to show short-wave KH instabilities.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

maxP2 / maxP

1 = 0.017044 maxP

2 / maxP

1 = 0.047946

maxP2 / maxP

1 = 0.015975 maxP

2 / maxP

1 = 0.35601

Figure: Typical unstable modes(left to right, top to bottom):KF1, RF, RP, PF.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Scenario of development of the baroclinic RFinstability as follows from DNS

1. Upper layer: frontal wave evolves into a series ofmonopolar vortices at certain spacing due to vortexlines clipping and reconnection following formation ofKelvin fronts

2. Lower layer: Rossby wave develops a series ofvortices of alternating signs

3. Lower-layer dipoles drive the vortex out of the shoreand are at the origin of the detachment.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

y

x

t= 160

0 1 2 3 4 5 6−1

0

1

2

y

x

t= 160

0 1 2 3 4 5 6−1

0

1

2

y

x

t= 200

0 1 2 3 4 5 6−1

0

1

2

y

x

t= 200

0 1 2 3 4 5 6−1

0

1

2

Figure: Levels of h1(x , y , t) in the upper layer (left) and isobarsof π2(x , y , t) in the lower layer (right) at t = 150 and 200 for thedevelopment of the unstable RF mode superposed on thebasic flow with a depth ratio r = 2.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Energetics

0 50 100 150 200 250 300

−8

−7

−6

−5

−4

−3

−2

−1

0

t

log

(Kp

er)

Figure: Logarithm of the kinetic energy Kper of the perturbationfor the unstable mode in the upper layer (thick) and in the lowerlayer (dashed).



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Kelvin front and dissipation duringdevelopment of RF instability

y

x

t= 155

3 3.5 4 4.5 5 5.5 6−1

−0.5

Figure: Before detachment: zoom of the wall region.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Structure of the detached vortex 1

y

x

t= 225

2 3 4 5 6 7

0

1

2

3

4

Figure: Isobars of π1(x , y , t) in the upper layer (white lines) andπ2(x , y , t) in the lower layer (dark lines) at t = 250. Dark (light)background: anticyclonic (cyclonic) region.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Structure of the detached vortex 2

3 4 5 6 7 8 90

0.5

1

1.5

2

2.5

h

x0 1 2 3 4 5 6

0

0.5

1

1.5

2

2.5

h

y

Figure: The x (left) and y (right) cross-sections of the detachedvortex at t = 300



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Evolution of the total energy

0 50 100 150 200 250 3000.9996

0.9997

0.9998

0.9999

1

t

E/E0



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

y

x

t= 8

0 1 2 3 4−1

0

y

x

t= 8

0 1 2 3 4−1

0

y

x

t= 60

0 1 2 3 4−1

0

1

2

y

x

t= 60

0 1 2 3 4−1

0

1

2

Figure: Levels of h1(x , y , t) in the upper layer (left) and isobarsof π2(x , y , t) in the lower layer (right) at t = 20 and 60 for thedevelopment of the unstable RF mode superposed on thebasic flow with a depth ratio r = 0.5.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Energetics

0 20 40 60 80 100 120−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

t

log

(Kp

er)

0 20 40 60 80 100 120

0.998

0.9985

0.999

0.9995

1

t

E/E0

Figure: Left -logarithm of the kinetic energy of the perturbationfor mode k = k0 in the upper layer (solid) and in the lower layer(dashed), and for the sum of modes with k > 10 k0(dashed-dotted ). Right - time-dependence of the total energy(thick line) and the dissipation rate (dashed line) for theevolution of the instability



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

Loss of hyperbolicity

y

0.5 1 1.5 2 2.5 3 3.5 4−1

0

−6

−4

−2

0

2

4x 10

−3

x

y

0 1 2 3 4−1

0

−5

0

5

x 10−3

Figure: Contours of π1(x , y , t) (upper panel) and π2(x , y , t)(lower panel) with mean zonal flow filtered out at t = 20, with adepth ratio r = 0.5. The white lines indicate the boundaries ofnon-hyperbolic domains.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

I Exhaustive linear stability analysis of coastalcurrents/oucropping fronts performed,

I Physical nature of all instabilities as resonancesbetween various eigenmodes established,

I Nonlinear evolution of leading instabilities simulatedwith new high-resolution finite-volume code,

I An essential role of Kelvin fronts (breaking Kelvinwaves) in reorganization of the flow and coherentstructure formation highlighted,

I A mechanism of vortex detachment from theunstable baroclinic coastal current is identified,

I (Non-) Influence of short-scale shear instabilitiesunderstood.



coastal currents

Motivation

Approach


Linear stability




Linear stability




Literature

References

Presentation is based onI J. Gula and V. Zeitlin, "Instabilities of

buoyancy-driven coastal currents and their nonlinearevolution in the two-layer rotating shallow watermodel. Part 1. Passive lower layer", J. Fluid Mech.,659, 69 - 93 (2010).

I J. Gula, V. Zeitlin and F. Bouchut, "Instabilities ofbuoyancy-driven coastal currents and their nonlinearevolution in the two-layer rotating shallow watermodel. Part 2. Active lower layer" J. Fluid Mech.,665, 209 - 237 (2010).

MotivationApproachPassive lower layerPreliminariesLinear stabilityNonlinear saturationManifestation of the instability in initial-value problem

Active lower layerPreliminariesLinear stabilityNonlinear saturationRole of the vertical shear (KH) instability

Summary and conclusionsLiterature

understanding large-scale atmospheric and oceanic flows ... · v. zeitlin, non-homogeneous fluids...

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