Slow dynamics in the fast rotation, order one stratification
limit.
Beth A. Wingate
Collaborators:
Miranda Holmes:
New York University Courant Institute
Mark Taylor:
Sandia National Laboratories
Pedro Embid:
University of New Mexico
Image courtesy NOAA, 2007
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High rotation rate effects - Taylor Columns
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Taylor-Proudman theorem: In
rapidly rotating flow, the flow twodimensionalizes.
Hogg, “On the stratified Taylor
Column” JFM, 1973
Davies, “Experiments on Taylor
Columns in Rotating and
Stratified Flow, JFM 1971
Taylor columns have been
observed in the high latitudes,
and they frequently involve some
degree of stratification too. These
are called Taylor Caps. See for
example Mohn, Bartsch, Meinche
Journal of Marine Science, 2002
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Triply periodic rotating and stratified Boussinesq equations
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Method of Multiple Scale -- Embid and Majda style.
To avoid secularity the second order term must be smaller than
the leading order term.
Embid, P and Majda, A “Low Froude number limiting dynamics for
stably stratified flow with small or finite Rossby number.”
Geophysical and Astrophysical Fluid Dynamics, 87 pp 1-50
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The equations for the slow dynamics are,
u, v, w all functions of (x,y) only.
,
Note change of notation.
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Numerical Simulations
High wave number white noise forcing. Smith and Waleffe,
JFM, 2002
Velocity
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Numerical Simulations
High wave number white
noise forcing. Smith and
Waleffe, JFM, 2002
Velocity
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Numerical simulations - Globally integrated total and slow
potential enstrophy.
High wave number white
noise forcing. Smith and
Waleffe, JFM, 2002
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Take home message
1. Following Embid and Majda we use the method of
multiple scales and fast wave averaging to find slow
dynamics for Rossby to zero, finite Froude limit.
2. The leading order solution to the stably stratified, triply
periodic fast-rotation Boussinesq equations has both
fast and slow components. Leading order potential
enstrophy is slow.
3. The slow dynamics evolves independently of the fast.
4. Equations for the slow dynamics including their
conservation laws. Two-D NS and w/theta dynamics.
5. Preliminary numerical results support the slow
conservation laws and the potential enstrophy only
slow when Ro is small.
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The Nondimensional Boussineseq Equations
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Boussinesq equations In non-local form
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In operator form
Embid and Majda, 1996, 1997
Schochet, 1994
Babin, Mahalov, Nicolaenko, 1996
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Method of Multiple scales - write the abstract form with epsilon
and tau.
To lowest order
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Method of multiple of time scales
The order 1solution is a function of the leading order
solution.
Duhammel’s formula
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Separation of time scales - Fast wave averaging equation
Where
solves:
And therefore, the solution to leading order is
And since the fast linear operator is skew-Hermitian,
has an
orthogonal decomposition into fast and slow components:
Using the fact that
like,
, the leading order solution looks
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Separation of time scales - Fast wave averaging equation
Therefore, study the solutions of the linear problem:
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By direct computation the fast wave averaging equation
Compute the evolution equation for the Fourier amplitudes
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Example of a linear operator
For the linear operators the only contributions from averaging in
time are the result of modes with the same frequency and wave
number. So they only contribute slow modes to the evolution of the
slow mode dynamics.
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The nonlinear operator
Three wave resonances means we must choose k’
and k’’ such that
You can show that the interaction coefficients are
zero for the fast-fast interaction. Which means that
the slow dynamics evolves independently of the fast.
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Find the equations for the slow dynamics
Knowing the slow dynamics evolves independently of the
fast we can find the equations for the slow dynamics by
projecting the solution and the equations onto the null
space of the fast operator
.Then the fast wave
averaged equation for the slow modes becomes,
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The null space of the fast operator
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The leading order conservation laws in the absence of
dissipation,
Using the same arguments as Embid and Majda, 1997,
there is conservation in time of the slow to fast energy
ratio. This means the total energy conservation is
composed of both slow and fast dynamics.
But the leading order potential enstrophy is composed
only of the slow dynamics.
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Analysis of the fast operator
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Do the horizontal and vertical kinetic energies decouple?
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