Geoscience Reference
In-Depth Information
where
f
is a suitable smooth function that defines a
Poincaré plane of intersection. With this method, integra-
tion over time
τ
is necessary for each GMRES iteration.
Unfortunately
τ
is often much higher than the time
t
opti-
mized for GMRES. This makes the numerical solution of
(2.26) very costly.
By using the hypotheses that the periodic solutions are
waves traveling with constant phase speed
ω
and these
waves do not deform (e.g., the fixed points of types 2
and 3 discussed in Section 2.4), the time consumption is
rendered comparable to that of the method for finding
stationary solutions described above. Instead of (2.26),
we solve
computation of the stability of the solutions, the transi-
tions between the various complex flows can be found.
This will lead to a better understanding of how the insta-
bilities lead to the observed dynamics, which in turn will
shed light on the dynamics of all differentially heated
rotating fluids.
Acknowledgments.
The author would like to acknowl-
edge the support of the Natural Sciences and Engineering
Research Council of Canada and SHARCNET.
REFERENCES
(
X
i
+1
,
t
,
α
i
+1
)
−
R
(ω
i
+1
t)
X
i
+1
=0,
Castrejon-Pita, A., and P. Read (2007), Baroclinic waves in
an air-filled thermally driven rotating annulus,
Phys. Rev.
E
,
75
(2), 026301. doi: http://dx.doi.org/10.1103/PhysRevE.
75.026301.
Fein, J. (1973), An experimental study of the effects of the upper
boundary condition on the thermal convection in a rotating
cylindrical annulus of water,
Geophys. Fluid Dyn.
,
5
, 213-248.
Fein, J., and R. Pfeffer (1976), Experimental study of effects of
Prandtl number on thermal convection in a rotating, differen-
tially heated cylindrical annulus of fluid,
J. Fluid Mech.
,
75
,
81-112.
Fowlis, W., and R. Hide (1965), Thermal convection in a rotating
annulus of liquid: Effect of viscosity on the transition between
axisymmetric and non-axisymmetric flow regimes,
J. Atmos.
Sci.
,
22
, 541-558.
Früh, W.-G., and P. Read (1997), Wave interactions and the
transition to chaos of baroclinic waves in a thermally driven
rotating annulus,
Philos. Trans. R. Soc. London, A
,
355
,
101-153.
Govaerts, W. (2000),
Numerical Methods for Bifurcations of
Dynamical Equilibria
, SIAM, Philadelphia, Penna.
Guckenheimer, J., and P. Holmes (1983),
Nonlinear Oscillations,
Dynamical Systems and Bifurcations of Vector Fields
,
Applied
Mathematical Sciences
, vol. 42, Springer-Verlag, New York.
Hassard, B., N. Kazarinoff, and Y.-H. Wan (1981),
Theory
and Applications of Hopf Bifurcation
,
London Mathematical
Society Lecture Note Series
, vol. 41, Cambridge Univ. Press,
Cambridge, UK.
Hennessy, M., and G. Lewis (2013), The primary flow transition
in a differentially heated rotating channel of fluid with O(2)
symmetry,
J. Computat. Appl. Math.
,
254
, 116-131.
Henry, D. (1981),
Geometric Theory of Semilinear Parabolic
Equations
,
Lecture Notes in Mathematics
, vol. 840, Springer-
Verlag, Berlin.
Hide, R., and J. Mason (1975), Sloping convection in a rotating
fluid,
Adv. Geophys.
,
24
, 47-100.
Hignett, P. (1985), Characteristics of amplitude vacillation in
a differentially heated rotating fluid annulus,
Geophys. Astro-
phys. Fluid Dynam.
,
31
, 247-281.
Hignett, P., A. White, R. Carter, W. Jackson, and R. Small
(1985), A comparison of laboratory measurements and
numerical simulations of baroclinic wave flows in a rotating
cylindrical annulus,
Q. J. R. Meteorol. Soc.
,
111
, 131-154.
X
i
+1
)
α
i
+1
)t
iα
(
X
i
+1
−
·
t
i
X
+
(α
i
+1
−
ω
i
+1
)t
iω
= 0,
+
(ω
i
+1
−
(2.27)
X
i
+1
)
(
X
i
+1
−
·
r
=0.
Here,
R
(ωt)
X
i
+1
represents the discretization of the veloc-
ity and temperature fields rotated over an angle
ωt
in the
azimuthal direction and
r
is the generator of these rota-
tions for
X
i
+1
. The phase speed
ω
replaces the period
τ
in the set of unknowns. The third equation imposes the
condition that the consecutive Newton-Raphson update
steps do not shift the azimuthal phase of the traveling
wave. The rotated fields are efficiently computed in Fourier
space as an element-wise multiplication by phase factors.
Using forward and backward fast Fourier transforms for
the angular coordinate
φ
, this is done at a computational
cost that is negligible compared to the time stepping.
The combination of pseudo-arclength continuation
with GMRES was first introduced by
Sánchez et al.
[2004]
and is called
Newton-Krylov
continuation.
Viswanath
[2007] later adapted the method to the computation of
equilibrium and modulated traveling waves. A third appli-
cation worth mentioning here, in the light of the normal
form analysis presented in Section 2.4, is that to the com-
putation of invariant tori.
Sánchez et al.
[2010] cast this
problem into a form suitable for Newton-Krylov con-
tinuation. The computation of quasi-periodic solutions
is much less straightforward than that of equilibria and
periodic solutions, though, and might fail if the quasi-
periodic solution is strongly unstable or has a high ratio
of frequencies. Nevertheless, it could be applied to exam-
ine quasi-periodic flows in regions of the parameter space
where stable fixed points of type 4 exist or regions further
from the transition where amplitude or other vacillation
is observed.
Armed with these methods, a detailed characteriza-
tion of the parameter space within the nonaxisymmetric
regime can be determined. In particular, coupled with the
