40 research outputs found
Navier-Stokes equations on a rapidly rotating sphere
We extend our earlier β-plane results [al-Jaboori and Wirosoetisno, 2011, DCDS-B 16:687--701] to a rotating sphere. Specifically, we show that the solution of the Navier--Stokes equations on a sphere rotating with angular velocity 1/ϵ becomes zonal in the long time limit, in the sense that the non-zonal component of the energy becomes bounded by ϵM. Central to our proof is controlling the behaviour of the nonlinear term near resonances. We also show that the global attractor reduces to a single stable steady state when the rotation is fast enough
Slow manifolds and invariant sets of the primitive equations
The authors review, in a geophysical setting, several recent mathematical results on the forced–dissipative hydrostatic primitive equations with a linear equation of state in the limit of strong rotation and stratification, starting with existence and regularity (smoothness) results and describing their implications for the long-time behavior of the solution. These results are used to show how the solution of the primitive equations in a periodic box comes close to geostrophic balance as t → ∞. Then a review follows of how geostrophic balance could be extended to higher orders in the Rossby number, and it is shown that the solution of the primitive equations also satisfies a higher-order balance up to an exponentially small error. Finally, the connection between balance dynamics in the primitive equations and its global attractor, which is the only known invariant set (for a sufficiently general forcing), is discussed
Semi-geostrophic particle motion and exponentially accurate normal forms
We give an exponentially-accurate normal form for a Lagrangian particle
moving in a rotating shallow-water system in the semi-geostrophic limit, which
describes the motion in the region of an exponentially-accurate slow manifold
(a region of phase space for which dynamics on the fast scale are exponentially
small in the Rossby number). The result extends to numerical solutions of this
problem via backward error analysis, and extends to the Hamiltonian
Particle-Mesh (HPM) method for the shallow-water equations where the result
shows that HPM stays close to balance for exponentially-long times in the
semi-geostrophic limit. We show how this result is related to the variational
asymptotics approach of [Oliver, 2005]; the difference being that on the
Hamiltonian side it is possible to obtain strong bounds on the growth of fast
motion away from (but near to) the slow manifold
Timestepping schemes for the 3d Navier-Stokes equations
It is well known that the (exact) solutions of the 3d Navier–Stokes equations remain bounded for all time if the initial data and the forcing are sufficiently small relative to the viscosity. They also remain bounded for a finite time for arbitrary initial data in L2. In this article, we consider two temporal discretisations (semi-implicit and fully implicit) of the 3d Navier–Stokes equations in a periodic domain and prove that their solutions remain uniformly bounded in H1 subject to essentially the same respective smallness conditions as the continuous system (on initial data and forcing or on the time of existence) provided the time step is small
Renormalization group method applied to the primitive equations
AbstractIn this article we study the limit, as the Rossby number ε goes to zero, of the primitive equations of the atmosphere and the ocean. From the mathematical viewpoint we study the averaging of a penalization problem displaying oscillations generated by an antisymmetric operator and by the presence of two time scales
Averaging method applied to the three-dimensional primitive equations
In this article we study the small Rossby number asymptotics for the three-dimensional primitive equations of the oceans and of the atmosphere. The fast oscillations present in the exact solution are eliminated using an averaging method, the so-called renormalisation group method
Some spectral applications of McMullen's Hausdorff dimension algorithm
Using McMullen's Hausdorff dimension algorithm, we study numerically the dimension of the limit set of groups generated by reflections along three geodesics on the hyperbolic plane. Varying these geodesics, we found four minima in the two-dimensional parameter space, leading to a rigorous result why this must be so. Extending the algorithm to compute the limit measure and its moments, we study orthogonal polynomials on the unit circle associated with this measure. Several numerical observations on certain coefficients related to these moments and on the zeros of the polynomials are discussed. - See more at: http://www.ams.org/journals/ecgd/2012-16-10/S1088-4173-2012-00244-5/home.html#sthash.MXrRFUVZ.dpu
The Euro Diffusion Project
From 1st January 2002 we have the unique possibility to follow the spread of national euro coins over the different European countries. We model and analyse this movement and estimate the time it will take before on average half the coins in our wallet will be foreign
Exponentially accurate balance dynamics
By explicitly bounding the growth of terms in a singular perturbation expansion with a small parameter ", we show that it is possible to find a solution that satises a balance relation (which denes the slow manifold) up to an error that scales exponentially in " as " ! 0. This is rst done for a generic nite-dimensional dynamical system with polynomial nonlinearity, followed by a continuous
uid case. In addition, for the finite-dimensional system, we show that, properly initialised, the solution of the full model stays within an exponential distance to that of the balance equation (i.e. evolution on the slow manifold) over a timescale of order one (independent of ")