Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Casimir preserving scheme for long-time simulation of spherical ideal
  hydrodynamics   The incompressible 2D Euler equations on a sphere constitute a fundamental
model in hydrodynamics. The long-time behaviour of solutions is largely
unknown; statistical mechanics predicts a steady vorticity configuration, but
detailed numerical results in the literature contradict this theory, yielding
instead persistent unsteadiness. Such numerical results were obtained using
artificial hyperviscosity to account for the cascade of enstrophy into smaller
scales. Hyperviscosity, however, destroys the underlying geometry of the phase
flow (such as conservation of Casimir functions), and therefore might affect
the qualitative long-time behaviour. Here we develop an efficient numerical
method for long-time simulations that preserve the geometric features of the
exact flow, in particular conservation of Casimirs. Long-time simulations on a
non-rotating sphere then reveal three possible outcomes for generic initial
conditions: the formation of either 2, 3, or 4 coherent vortex structures.
These numerical results contradict the statistical mechanics theory and show
that previous numerical results, suggesting 4 coherent vortex structures as the
generic behaviour, display only a special case. Through integrability theory
for point vortex dynamics on the sphere we present a theoretical model which
describes the mechanism by which the three observed regimes appear. We show
that there is a correlation between a first integral $\gamma$ (the ratio of
total angular momentum and the square root of enstrophy) and the long-time
behaviour: $\gamma$ small, intermediate, and large yields most likely 4, 3, or
2 coherent vortex formations. Our findings thus suggest that the likely
long-time behaviour can be predicted from the first integral $\gamma$.

28
The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
26
The categories of this paper are: bayes-an
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
17
The categories of this paper are: q-fin
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
30
The categories of this paper are: q-alg
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol