# Research accomplishments and interests

My research interests lie in applied mathematics, geophysical fluid dynamics, and in particular, the dynamics of the coastal ocean. Simplified dynamical models and numerical simulations are used to pursue fundamental understanding of how the oceans move in the way they do.

Contents1. Recent contributions

a) Exact solutions in a nonlinear theory of upwelling

b) Topographic effects in upwelling

2. Previous research topics

a) Baroclinic dynamics of abyssal gravity-driven flows

b) Abyssal equator-crossing currents

c) Atmospheric Rossby waves

3. Current projects

a) Exact solutions of upwelling and downwelling over sloping topography

b) Cross-shore velocity structure of deep shoreward flow

c) Efficient assimilation of AUV data into a numerical ocean model

4. Future research directions

a) 3D theory of coastal circulation, California Undercurrent

b) Data-assimilating numerical ocean models

**1. Recent contributions**

Much of my recent research has been focused on the dynamics of wind-driven coastal upwelling, a fundamental process in the large-scale circulation of the coastal ocean. Southward winds off the west coast of North America push the surface waters offshore, thereby drawing cool, nutrient-rich water to the surface from below. The drawing of nutrients to the sunlit surface drives the rich biological production in coastal regions. The biological importance of the upwelling process has motivated decades of study of the dynamics of the upwelling process itself (Allen 1980; Brink 1991; Hickey 1998). An understanding of the cross-shore and alongshore transport in the coastal ocean is also motivated by, for example, pollution transport, harmful algal blooms, and sediment transport. The same upwelling that fisheries rely upon for their success can transport oxygen-depleted water onto the continental shelf and cause mass die-offs of fish and invertebrates (Grantham et al. 2004).

**(a) Exact solutions of a nonlinear theory of upwelling**

Pedlosky (1978a,b) derived an elegant model of upwelling that expressed the cross-slope velocity, the alongslope velocity and the density of the ocean water as solutions to simple partial differential equations. The model was originally solved in each of two limiting cases: at steady state and as an initial-value problem. However, the solution of the initial-value problem is inconsistent with the steady-state solution. Roger Samelson, John Allen and I found a new solution to the initial-value problem that is consistent with the existing steady-state solution (Choboter, Samelson, and Allen, 2005). The new solution was found using coordinate transformations and by modifying the boundary conditions of the problem. The new solution gives insight into the dynamics of upwelling by making specific predictions about the time evolution of the density and velocity, given the dynamical assumptions that went into the Pedlosky upwelling model.

**(b) Topographic effects in upwelling**

The exact solution described above was found under the assumption that the ocean was uniform depth. However, the Pedlosky model applies even when the cross-shore topography depends on distance from shore. In this research, John Allen, Roger Samelson and I (manuscript in preparation) show that over a class of topographic profiles, including profiles with a continental shelf break, the model predicts high density gradients and onshore velocities near the ocean bottom over the shelf break. In fact, to solve the steady-state problem, the bottom boundary conditions must be modified. Mathematically, the problem resembles a type of free-boundary problem, called an obstacle problem, where the location of the domain boundary itself needs to be found as part of the solution. Exactly which topographic profiles lead to these solutions is documented, and the solutions are studied analytically and numerically. A physical explanation of the relation between vertical shear of onshore flow and the shape of the bottom topography is given in terms of streamline curvature and maintenance of the thermal wind relation. Thus, while bottom-intensification of onshore flow is known to occur via frictional bottom layers, the implication of this work is that such bottom intensification can occur due to inviscid processes.

**2. Previous research topics (performed while in graduate school)**

**(a) Baroclinic dynamics of abyssal gravity-driven flows**

Gordon Swaters and I studied the dynamics of density-driven and topographically steered flow using a reduced-dynamics model. This type of flow occurs in the abyssal ocean, as part of the global thermohaline circulation. In preparation to study similar dynamics in a laboratory setting (with a cylindrical rotating tank), the model was recast into a cylindrical coordinates and the stability of an annulus-shaped current was studied (Choboter and Swaters, 2000). As part of the analysis, we studied how effectively the predictions of this model compares with experimental data taken by other researchers. We showed that the laboratory data collected was consistent with the predictions of the model in certain parameter ranges.

**(b) Abyssal equator-crossing currents**

Large-scale density-driven ocean flow may be largely understood by inviscid dynamics: the conservation of potential vorticity is an accurate approximation under many circumstances. However, part of the global thermohaline circulation is observed to undergo order-one changes in its potential vorticity as it crosses the equator. This motivated my Ph.D. thesis studying the dynamics of abyssal equator-crossing flows. I derived a reduced-dynamics model of these flows, which is simpler than the full equations of motion, but still retains the effects of friction and bottom topography on the fluid (Choboter and Swaters, 2003). Analytical work on this model suggests that fluid will be in geostrophic balance to first order when far from the equator, and at the equator the fluid will flow down the pressure gradient, maintaining a balance between friction and the pressure gradient. Numerical simulations of this model were compared to simulations of a more complete set of equations of motion, the shallow water equations. It was shown that the simple model, while neglecting the inertia of the fluid, does capture some of the features of the flow quite well.

The flow of Antarctic Bottom Water (AABW) along the ocean bottom as it approaches and crosses the equator was numerically simulated using a reduced-gravity shallow water model (Choboter and Swaters, 2004). Analysis of these simulations highlighted the role that nonlinear advection plays in the dynamics, in particular near certain topographic features. We found an interesting dynamical feature of these flows: a topographic basin located at the equator interacts with the dense fluid to temporarily store it in the basin, amplifying a weak temporal variation in the volume flux. This research suggested that the large annual variation in flux of AABW crossing the equator seen in observational data may be a consequence of a relatively small variation in the flux of AABW at the source, far south of the equator.

**(c) Atmospheric Rossby waves**

Rossby waves, which are a certain class of large-scale atmospheric waves, require a background potential vorticity gradient to propagate. In regions where the potential vorticity gradient vanishes, the propagation is disrupted, and energy may build. There are regions of the atmosphere where eddy stirring tends to eliminate the potential vorticity gradient, but in the same locations are also sources of energy that may drive Rossby wave production. In this research, Gilbert Brunet, Sherwin Maslowe and I studied the evolution of forced Rossby waves in a zero ambient potential vorticity field (Choboter, Brunet, and Maslowe, 2000). Analysis of the linearized problem showed how waves would increase in amplitude as a result of the forcing. The numerical solution of the nonlinear problem demonstrated that the nonlinearity and the forcing terms acted in concert to halt the growth of the waves, until a combination of stationary and steadily-propagating vortices were superimposed at quasi-steady state. Thus, the presence of forcing does not lead to unlimited growth of vorticity, but instead the system tends toward a state where the forcing balances with the nonlinear propagation of vorticity.

**3. Current projects**

**(a) Exact solutions of upwelling and downwelling over sloping topography**

The exact solution to the Pedlosky upwelling model described flow over a constant-depth ocean. However, with a change of variables, the exact solution over constant-depth ocean can be transformed into a new solution over variable-depth ocean for specific functional forms of the topography, and also transformed into a solution that describes the downwelling process (where the winds and ﬂuid paths reverse direction). In the summer of 2009, I worked with Cal Poly mathematics undergraduate students Dana Duke, JP Horton and Paul Sinz, to explore these new solutions, and to study how the evolution of the velocity and ocean density depend on bottom slope, particularly in the downwelling case. We investigated the analytical solutions and compared these to more realistic simulations using the Princeton Ocean Model. The analytic solution captures well certain aspects of downwelling seen in the numerical simulations.

**(b) Cross-shore velocity structure of deep shoreward flow**

A two-dimensional theory of wind-driven coastal upwelling is developed that is comprised of a surface Ekman layer, an interior frictionless layer, and a frictional bottom boundary layer. The theory is built upon the Lentz-Chapman upwelling theory, which has been used to demonstrate the importance of nonlinear cross-shelf momentum flux divergence during upwelling. The new model retains spatially-varying structure in the interior density and velocity fields. The dynamical model for the interior flow is based upon the nonlinear upwelling theory of Pedlosky, which maintains thermal wind balance between the cross-shelf density gradient and the vertical shear in the alongshelf velocity while retaining the cross-shelf advection of density and alongshelf momentum. The structure of the cross-shelf circulation is studied as a function of alongshelf wind stress and Burger number S = α N/f, where α is the topographic slope, N is the buoyancy frequency, and f is the Coriolis parameter. Predictions of the dynamical model are compared with two-dimensional numerical model simulations. During upwelling winds, the dynamical model predicts interior onshore flow high in the water column for large Burger number, and onshore flow in the bottom boundary layer for small Burger number, consistent with the numerical model and with observations.

**(c) Efficient assimilation of AUV data into a numerical ocean model**

Recent research has identified how a data assimilation system may be used not only to improve the accuracy of a numerical model's forecast, but also to measure the sensitivity of the improvement with respect to the observations assimilated. The observation sensitivity is used in this work to predict optimal AUV paths for efficiently collecting data to be assimilated into a coastal ocean model. Numerical experiments are performed using the Regional Ocean Modeling System (ROMS), along with its tangent-linear and adjoint tools, to determine strategies for AUV path optimization. Simulation parameters, including forcing and boundary conditions, are chosen in preparation for runs assimilating AUV and CODAR data off the coast of central California. This is work in collaboration with Chris Clark of Computer Science and Mark Moline of Biological Sciences.

**4. Future research directions**

**(a) 3D theory of coastal circulation and the California Undercurrent**

One future research theme I intend to explore is the development of the theory of three-dimensional movement of coastal waters. The upwelling theory I have been working with recently has assumed alongshore uniformity, and thus is a two-dimensional (cross-slope and depth) model. Extending that model to include alongshore dependence of the variables would have to incorporate northward-propagating coastal trapped waves, the development of alongshore pressure gradients, and the California Undercurrent. The California Undercurrent (CUC) is a sub-surface poleward current located off the west coast of North America. The strength of the CUC varies in time and depends on location, but some signature of the northward-flowing CUC is present year-round, despite the fact that the winds and resulting surface currents are southward much of the year. The dynamical reasons for the presence of the CUC are poorly understood. The CUC may be one of a global family of currents with the same dynamical causes, since similar currents have been observed along the eastern boundary of each of the world's oceans and in both hemispheres, always flowing away from the equator.

**(b) Data-assimilating numerical ocean models**

There have been fantastic advances in recent years, both in the emergence of data-assimilating numerical models, and in the availability of oceanographic data. Community ocean models such as the Regional Ocean Modeling System (ROMS, http://www.myroms.org/) have developed into sophisticated research tools that include a suite of data assimilation capabilities. I have begun to use these capabilities in my research, and I am interested in developing ways of using these assimilation tools to advance the understanding of geophysical fluid dynamics of the coastal ocean. These numerical tools will also be particularly useful in my future interdisciplinary collaborations, as well as in research projects with students.

**References**

Allen, J. S. 1980 Models of wind-driven currents on the continental shelf. Ann. Rev. Fluid Mech. 12, pp. 389-433.

Brink 1991 Coastal-trapped waves and wind-driven currents over the continental shelf. Ann. Rev. Fluid Mech. 23, pp. 389-412.

Choboter, P. F., Samelson, R. M., and Allen, J. S. 2005 A new solution of a nonlinear model of upwelling. Journal of Physical Oceanography 35, no. 4, pp. 532-544.

Choboter, P. F. and Swaters, G. E. 2004 Shallow-water modeling of Antarctic Bottom Water crossing the equator. Journal of Geophysical Research - Oceans 109, no. C3, C03038, doi: 10.1029/2003JC002048.

Choboter, P. F. and Swaters, G. E. 2003 Two-layer models of abyssal equator-crossing flow. Journal of Physical Oceanography 33, pp. 1401-1415.

Choboter, P. F. and Swaters, G. E. 2000 On the baroclinic instability of axisymmetric rotating gravity currents with bottom slope. J. Fluid Mechanics 408, pp. 149-177.

Choboter, P. F., Brunet, G. and Maslowe, S. A. 2000 Forced disturbances in a zero absolute vorticity gradient environment. J. Atmospheric Sciences 57, no. 9, pp. 1406-1419.

Grantham, B. A., F. Chan, K. J. Nielsen, D. S. Fox, J. A. Barth, A. Huyer, J. Lubchenco, and B. A. Menge 2004 Upwelling-driven nearshore hypoxia signals ecosystem and oceanographic changes in the northeast Pacific. Nature, 429, 749-754, doi:10.1038/nature02 605.

Hickey, B. M. 1998 Coastal oceanography of western North America from the tip of Baja California to Vancouver Island. The Sea, Robinson, A. R. and K. H. Brink, Eds., John Wiley & Sons, Inc., Vol. 11, 345-393.

Pedlosky, J. 1978a An inertial model of steady coastal upwelling. J. Phys. Oceanogr., 8, 171-177.

Pedlosky, J. 1978b A nonlinear model of the onset of upwelling. J. Phys. Oceanogr., 8, 178-187.