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

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

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

A mollified Ensemble Kalman filter

It is well recognized that discontinuous analysis increments of sequential data assimilation systems, such as ensemble Kalman filters, might lead to spurious high frequency adjustment processes in the model dynamics. Various methods have been devised to continuously spread out the analysis increments over a fixed time interval centered about analysis time. Among these techniques are nudging and incremental analysis updates (IAU). Here we propose another alternative, which may be viewed as a hybrid of nudging and IAU and which arises naturally from a recently proposed continuous formulation of the ensemble Kalman analysis step. A new slow-fast extension of the popular Lorenz-96 model is introduced to demonstrate the properties of the proposed mollified ensemble Kalman filter.Comment: 16 pages, 6 figures. Minor revisions, added algorithmic summary and extended appendi

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

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

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