Baroclinic instability variations
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Baroclinic Instability Variations. F. Javier Beron-Vera J. Fluid Mech., 352: 245 (1997) M. Josefina Olascoaga J. Geophys. Res., 104: 23,357 (1999) Pedro Miguel Ripa J. Fluid Mech . , 403: 1 (2000) J. Fluid Mech . , 428: 387 (2001) Rev. Mex. Fís. , in press. Outline. Background

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Baroclinic instability variations
Baroclinic Instability Variations

  • F. Javier Beron-Vera

    J. Fluid Mech., 352: 245(1997)

  • M. Josefina Olascoaga

    J. Geophys. Res., 104: 23,357(1999)

  • Pedro Miguel Ripa

    J. Fluid Mech., 403: 1(2000)

    J. Fluid Mech., 428: 387(2001)

    Rev. Mex. Fís., in press


Outline
Outline

  • Background

  • Generalizing Two-field models

  • Charney Numbers – Arnold-1

  • Low- and Short-wave cutoff – Arnold-2

  • Components “resonance”

  • Rossby Waves resonance

  • Bounds on the growth of perturbations


Eady (1949): β = 0, rigid horizontal boundaries

Blumsack & Gierasch (1972): β = 0, sloping bottom

Fukamachi et al. (1995):β = 0, free bottom, βT = 0 (topographic)

Beron-Vera (1997):β = 0, free bottom, βT 0

Lindzen (1994): β  0, but q uniform.

Phillips (1951): β  0, rigid horizontal boundaries

Bretherton (1966): β = 0, rigid sloping boundaries

Olascoaga (1999) : β  0, free bottom , βT 0


Basic Flow: Two Charney # (beta/shear)

Outside the wedge, a hamiltonian (“energy”) is H > 0: nonlinear stability (Arnold’s First Theorem)


Enter another variable: the perturbation wavenumber κ

All hamiltonians are sign independent for

κL(b,bT) < κ < κS(b,bT)

Notice the finite region for κL(b,bT) = 0


Normal Mode instability, for κL(b,bT) < κ < κS(b,bT)


Growth Rate along different directions in the (b,bT) plane (see color coded regions in slide 5)

  • If the advection of a q field by the other q field is arbitrarely neglected, these uncoupled components “resonate” along the blue curve. This “explains” the instability onset. (If shear  0 then b, bT  along b/bT = const.)

  • Maximum growth rate near b + bT  0


Resonance of Rossby Waves

The conditionb + bT  0 corresponds to the cancellation of both beta effects+ T  0: near resonance of true waves.This “explains” the maximum growth rate.


Bounding the wavy part of the perturbation qj,à la Shepherd, or the whole perturbation qj


Conclusions
Conclusions

  • Generalized Phillips-like or Eady-like model:

    • either two layers with constant density and variable potential vorticity orone layer with constant potential vorticity and variable boundary densities

    • free boundary and/or fixed topography

  • Necessary and sufficient stability conditions

  • “Resonance” of uncouple dynamical fields

  • At the cancellation of planetary and topographic beta effects:

    • resonance of Rossby waves

    • maximum growth rate

    • perturbation growth bounds are trivial


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