Eulerian/Lagrangian relationships in 2D turbulence plus a few informal comments on rain, convection, and moisture transport

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The barotropic vorticity equation (with free
surface)
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Barotropic Rossby waves (rigid lid)
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Barotropic Rossby waves (rigid lid)
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Barotropic Rossby waves (rigid lid)
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Rossby waves
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The 2D vorticity equation ( f plane, no
free-surface effects )
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In the absence of dissipation and forcing, 2D
barotropic flows conserve two quadratic
invariants energy and enstrophy
As a result, one has a direct enstrophy
cascade and an inverse energy cascade
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Two-dimensional turbulence the transfer
mechanism
As a result, one has a direct enstrophy
cascade and an inverse energy cascade
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Two-dimensional turbulence inertial ranges
As a result, one has a direct enstrophy
cascade and an inverse energy cascade
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Two-dimensional turbulence inertial ranges
As a result, one has a direct enstrophy
cascade and an inverse energy cascade
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Two-dimensional turbulence inertial ranges
k-5/3
log E(k)
k-3
E
Z
log k
As a result, one has a direct enstrophy
cascade and an inverse energy cascade
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Is this all ?
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Vortices form, interact, and dominate the
dynamics Vortices are localized, long-lived
concentrations of energy and enstrophy Coherent
structures
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Vortex studies Properties of individual
vortices (and their effect on tracer
transport) Processes of vortex formation Vortex
motion and interactions, evolution of the vortex
population Transport in vortex-dominated flows
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Coherent vortices in 2D turbulence
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Qualitative structure of a coherent vortex
z
(u2v2)/2
Q(s2-z2)/2
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The Okubo-Weiss parameter
z
u2v2
Qs2-z2
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The Okubo-Weiss field in 2D turbulence
z
u2v2
Qs2-z2
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The Okubo-Weiss field in 2D turbulence
z
u2v2
Qs2-z2
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Coherent vortices trap fluid particles for long
times (contrary to what happens with linear
waves)
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Motion of Lagrangian particles in 2D turbulence
Formally, a non-autonomous Hamiltonian
system with one degree of freedom
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The Lagrangian view
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Effect of individual vortices Strong
impermeability of the vortex edges to inward and
outward particle exchanges
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Example the stratospheric polar vortex
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Vortex formation Instability of vorticity
filaments Dressing of vorticity peaks But why
are vortices coherent ?
Qs2-z2
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Instability of vorticity filaments
z
Qs2-z2
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Existing vortices stabilize vorticity
filaments Effects of strain and adverse shear
z
Qs2-z2
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Processes of vortex formation and evolution in
freely-decaying turbulence Vortex formation
period Inhibition of vortex formation by
existing vortices
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Vortex interactions Mutual advection (elastic
interactions) Opposite-sign dipole formation
(mostly elastic) Same-sign vortex merging,
stripping, etc (strongly inelastic) 2 to 1, 2 to
1 plus another, .
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A model for vortex dynamics The (punctuated)
point-vortex model
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Beyond 2D Free-surface effects Dynamics on the
b-plane Role of stratification
z
Qs2-z2
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The discarded effects free surface
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The discarded effects dynamics on the b-plane
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Filtering fast modes The quasigeostrophic
approximation in stratified fluids
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The stratified QG potential vorticity equation
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Vortex merging and filamentation in 2D turbulence
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Vortex merging and filamentation in QG
turbulence role of the Green function
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