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2016
V
OLUME 61
JOURNAL OF THE ATMOSPHERIC SCIENCES
q
2004 American Meteorological Society
PoissonBracket Approach to the Construction of Energy and PotentialEnstrophy
Conserving Algorithms for the ShallowWater Equations
R
ICK SALMON
Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California
(Manuscript received 11 September 2003, in ﬁnal form 27 February 2004)
ABSTRACT
Arakawa and Lamb discovered a ﬁnitedifference approximation to the shallowwater equations that exactly
conserves ﬁnitedifference approximations to the energy and potential enstrophy of the ﬂuid. The Arakawa–
Lamb (AL) algorithm is a stunning and important achievement—stunning, because in the shallowwater case,
neither energy nor potential enstrophy is a simple quadratic, and important because the simultaneous conservation
of energy and potential enstrophy is known to prevent the spurious cascade of energy to high wavenumbers.
However, the method followed by AL is somewhat ad hoc, and it is difﬁcult to see how it might be generalized
to other systems.
In this paper, the AL algorithm is rederived and greatly generalized in a way that should permit still further
generalizations. Beginning with the Hamiltonian formulation of shallowwater dynamics, its two essential in
gredients—the Hamiltonian functional and the Poissonbracket operator—are replaced by ﬁnitedifference ap
proximations that maintain the desired conservation laws. Energy conservation is maintained if the discrete
Poisson bracket retains the antisymmetry property of the exact bracket, a trivial constraint. Potential enstrophy
is conserved if a set of otherwise arbitrary coefﬁcients is chosen in such a way that a very large quadratic form
contains only diagonal terms. Using a symbolic manipulation program to satisfy the potentialenstrophy con
straint, it is found that the energy and potentialenstrophyconserving schemes corresponding to a stencil of 25
grid points contain 22 free parameters. The AL scheme corresponds to the vanishing of all free parameters. No
parameter setting can increase the overall accuracy of the schemes beyond second order, but 19 of the free
parameters may be independently adjusted to yield a scheme with fourthorder accuracy in the vorticity equation.
1. Introduction
In a remarkable paper, Arakawa and Lamb (1981,
hereafter AL) discovered a ﬁnitedifference scheme for
the shallowwater equations that—apart from errors as
sociated with the ﬁnite time step—exactly conserves
ﬁnitedifference analogs of the energy and potential en
strophy of the ﬂow. The practical importance of retain
ing energy and enstrophyconservation laws in nu
merical models had been appreciated since the earlier
work of Arakawa (1966) on twodimensional, nondiv
ergent ﬂow; but whereas the derivation of Arakawa’s
Jacobian for nondivergent ﬂow may be transparently
understood in a variety of ways (see, e.g., Salmon and
Talley 1989), the AL algorithm for shallowwater dy
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This note was uploaded on 01/08/2012 for the course MPO 662 taught by Professor Iskandarani,m during the Spring '08 term at University of Miami.
 Spring '08
 Iskandarani,M

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