117137965
Stochastic forcing of quasi-geostrophic eddies
The basic theoretical framework is presented for the study of the stochastic atmospheric forcing of barotropic quasi-geostrophic oceanic eddies. The governing equation is the linearized potential vorticity equation with a stationary stochastic forcing function representing random fluctuations in the atmospheric windstress curl. The frequency autospectrum of the response and the frequency cross-spectrum between forcing and response are spatial convolutions of the frequency spectrum of the forcing with the Green’s function of the potential vorticity equation. Coherence maps have been calculated for simple geometries (meridional channel or infinite ocean) and simple forcing spectra (Gaussian or white in wavenumber space). They reproduce basic features of observed coherence maps. The major open question is whether the more complex structure of the observed maps is due to the more complex structure of the forcing spectrum or due to the processes neglected in the potential vorticity equation.
118540313
On the diffusion of momentum and mass by internal gravity waves
122312791
STOCHASTIC FORCING OF OCEANIC MOTIONS
Most oceanic motions are forced by stresses and heat and fresh water fluxes at the air-sea interface. When these forcing fields are assumed to be prescribed and stochastic and when the oceanic response is assumed to be linear then the forcing problem reduces to a set of stochastically forced linear oscillators. This set can in principle be decoupled. The asymptotic response of a linear oscillator to stationary random forcing is well understood and depends on whether the oscillator is stable, unstable or neutral. The explicit decoupling requires additional simplifying assumptions as exemplified by the stochastic forcing of surface gravity waves, internal gravity waves, Rossby waves and sea-surface temperature anomalies A powerful diagnostic tool is a coherence map which desribes the coherence between the oceanic response at one location and the atmospheric forcing at another location as a function of separation for different frequancies. A simple model of the stochastic forcing of barotropic Rossby waves by fluctuations in the atmospheric windstress reproduces basic features of observed coherence maps. The model expecially accounts for the qualitative changes that occur when different oceanic variables are considered or when the frequency is changed.
123589108
On simple semiabelian p-adic lie algebras
15736393
Kronecker conjugacy of polynomials
Let f, g ∈ Z[X] be non-constant polynomials with integral coefficients. In 1968 H. Davenport raised the question as to when the value sets f(Z) and g(Z) are the same modulo all but finitely many primes. The main progress until now is M. Fried’s result that f and g then differ by a linear substitution, provided that f is functionally indecomposable. We extend this result to polynomials f of composition length 2. Also, we study the analog when Z is replaced by the integers of a number field. The above number theoretic property translates to an equivalent property of subgroups of a finite group, known as Kronecker conjugacy, a weakening of conjugacy which has been studied by various authors under different assumptions and in other contexts. We also give a simplified and strengthened version of the Galois theoretic translation to finite groups.
59486970
Hilbert’s irreducibility theorem for prime degree and general polynomials
Letf (X, t)εℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt0εℤ such thatf (X, t0) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t0) is irreducible for all but finitely manyt0εℤ unless the curve given byf (X, t)=0 has infinitely many points (x0,t0) withx0εℚ,t0εℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others.