# Science

I am a physicist and specialize in geophysical fluid dynamics. Currently I do research at NWRA studying inertial oscillations and eddies.

## Highlights

Numerical Models – Movies and code for a number of geophysical fluid dynamics models, including the a quasigeostrophic eddy, forced-dissipative turbulence, an unstable shear flow, surface gravity waves, Poincare waves, linear internal waves, and many others.

Inertial Oscillations – A brief introduction to inertial oscillations, both theoretical and observational. This also includes a Matlab script for computing inertial paths, velocities, and periods from the exact solutions.

Quasigeostrophic Eddies – Movies of the sea-surface height as observed from satellite altimetry and the sea-surface height of a quasigeostrophic model attempting to reproduce the observations. This also include a number of animations from the monopole study, where a quasigeostrophic eddy was allowed to evolve in isolation.

## Awaiting Publication

J. Early, R. Samelson. Near-geostrophic approximations of the spherical shallow-water equations. Journal of Physical Oceanography. In preparation. (pdf)

We use a novel technique for evaluating scales of motion in order find an appropriate model for altimeter tracked mesoscale eddies. Starting from the spherical shallow water equations and assuming geostrophic dominance, we derive a potential vorticity conservation law in terms of all four non-dimensional parameters inherent in the equations while retaining the spherical geometry. The resulting equation reduces to other existing geostrophic theories, such as quasi-geostrophy and the Flierl-Petviashvili equation, by assuming a precise relationship between the non-dimensional parameters. However, by retaining freedom in the parameters we can determine at what scales the various theories remain valid. We find that a new extension to the FP equation is required to describe the mid-latitude mesoscale eddies.

## Publications

J. Early The forces of inertial oscillations. Quarterly Journal of the Royal Meteorological Society. In press.

By starting with a free particle and successively adding constraints, it is shown that the free motion of a particle constrained to the Earth’s surface is inertial, despite statements in the literature to the contrary, and that an observer on this particle would not measure a force tangent to the Earth’s surface. However, if the observer extended his measurements to include the direction normal to the Earth’s surface then he would detect an oscillating force.

D. Chelton, P. Gaube, M. Schlax, J. Early, R. Samelson. The influence of nonlinear mesoscale eddies on near-surface oceanic chlorophyll. Science. 2011. (doi)

Oceanic Rossby waves have been widely invoked as a mechanism for large-scale variability of chlorophyll (CHL) observed from satellites. High-resolution satellite altimeter measurements have recently revealed that sea-surface height (SSH) features previously interpreted as linear Rossby waves are nonlinear mesoscale coherent structures (referred to here as eddies). We analyze 10 years of measurements of these SSH fields and concurrent satellite measurements of upper-ocean CHL to show that these eddies exert a strong influence on the CHL field, thus requiring reassessment of the mechanism for the observed covariability of SSH and CHL. On time scales longer than 2 to 3 weeks, the dominant mechanism is shown to be eddy-induced horizontal advection of CHL by the rotational velocities of the eddies.

J. Early, R. Samelson, D. Chelton. Long-term evolution of quasi-geostrophic eddies. Journal of Physical Oceanography. August 2011.

The long-term evolution of initially Gaussian eddies is studied in a reduced-gravity shallow-water model using both linear and nonlinear quasi-geostrophic theory in an attempt to understand westward propagating mesoscale eddies observed and tracked by satellite altimetry. By examining both isolated eddies and a large basin seeded with eddies with statistical characteristics consistent with those of the observed eddies, it is shown that long term eddy coherence and the zonal wavenumber-frequency power spectral density are best matched by the nonlinear model. Individual characteristics of the eddies including amplitude decay, horizontal length scale decay, zonal and meridional propagation speed of a previously unrecognized quasi-stable state are examined. The results show that the meridional deflections from purely westward flow (poleward for cyclones and equatorward for anticyclones) are consistent with observations but that the limiting zonal propagation speed lacks the variability of satellite observations. Examination of the fluid transport properties of the eddies shows that an inner core of the eddy, defined by the zero relative vorticity contour, contains only fluid from the eddy origin, while a surrounding outer ring contains a mixture of  ambient fluid from throughout the eddy’s lifetime.

J. Early, J. Pohjanpelto, R. Samelson. Group foliation of equations in geophysical fluid dynamics. Discrete and Continuous Dynamical Systems, Series A. March 2010. (pdfdoi)

The method of group foliation can be used to construct solutions to a system of partial differential equations that, as opposed to Lie’s method of symmetry reduction, are not invariant under any symmetry of the equations. The classical approach is based on foliating the space of solutions into orbits of the given symmetry group action, resulting in rewriting the equations as a pair of systems, the so-called automorphic and resolvent systems, involving the differential invariants of the symmetry group, while a more modern approach utilizes a reduction process for an exterior differential system associated with the equations. In each method solutions to the reduced equations are then used to reconstruct solutions to the original equations. We present an application of the two techniques to the one-dimensional Korteweg-de Vries equation and the two-dimensional Flierl-Petviashvili (FP) equation. An exact analytical solution is found for the radial FP equation, although it does not appear to be of direct geophysical interest.

## Unpublished Notes

### 2011 – Stokes drift of Rossby waves (pdf)

Unpublished computation of the Stokes drift due to Rossby waves. A single plane wave does not induce stokes drift, but two plane waves can.