[Show abstract][Hide abstract] ABSTRACT: A time-dependent physical model whose initial condition is only approximately known can predict the evolving physical state to only within certain error bounds. In the prediction of weather, as well as its ocean counterpart, quantifying this uncertainty or the predictability is of critical importance because such quantitative knowledge is needed to provide limits on the forecast accuracy. Monte Carlo simulation, the accepted standard for uncertainty determination, is impractical to apply to the atmospheric and ocean models, particularly in an operational setting, because of these models' high degrees of freedom and computational demands. Instead, methods developed in the literature have relied on a limited ensemble of simulations, selected from initial errors that are likely to have grown the most at the forecast time. In this paper, the authors present an alternative approach, the polynomial chaos method, to the quantification of the growth of uncertainty. The method seeks to express the initial errors in functional form in terms of stochastic basis expansions and solve for the uncertainty growth directly from the equations of motion. It is shown via a Lorenz model that the resulting solution can provide all the error statistics as in Monte Carlo simulation but requires much less computation. Moreover, it is shown that the functional form of the solution facilitates the uncertainty analysis. This is discussed in detail for the tangent linear case of interest to ensemble forecasting. The relevance of the uncertainty covariance result to data assimilation is also noted.
Preview · Article · Jun 2010 · Journal of Atmospheric and Oceanic Technology
[Show abstract][Hide abstract] ABSTRACT: This paper deals with flow in a rectilinear channel on a rotating earth. The flow is directed perpendicular to the background planetary vorticity; both an analytical theory and numerical simulations are employed. The analytical approach assumes the existence of an eddy viscosity and employs a perturbation expansion in powers of the reciprocal of the Rossby number (Ro). At lowest order, a cross-channel circulation arises because of the tilting of the planetary vorticity vector by the shear in the along-channel direction. This circulation causes a surface convergence, which achieves its maximum value at a channel aspect ratio (= width/depth) of approximately 10. The location of the maximum surface convergence moves from near the center of the channel to a position very near the sidewalls as the aspect ratio increases from O(1) to O(100). To include the effects of turbulence, direct numerical pseudospectral simulations of the equations of motion are employed. While holding the friction Reynolds number fixed at 230.27, a series of simulations with increasing rotation (Ro = ∞, 10, 1.0, 0.1) are performed. The channelwide circulation cell observed in the analytical theory occurs for the finite Rossby number, but is displaced by lateral self-advection. In addition, turbulence-driven corner circulations appear, which make the along-channel maximum velocity appear at a subsurface location. The most interesting effect is the segregation of the turbulence to one side of the channel, while the turbulence is suppressed on the opposite side.
[Show abstract][Hide abstract] ABSTRACT: Internal solitary waves referred to as solitons are common occurrences
in the South China Sea. The Asian Seas International Acoustics
Experiment (ASIAEX) experiment was carried out in April-May 2001 to
perform measurements on these solitons, which are often highly
energetic, having isopycnal displacements well over 100 m and phase
speeds greater than 2.5 m/s. Of particular interest is the interaction
of a soliton with the sloping shelf bottom that occurs as the soliton
shoals to water depths less than its wave height. Observations during
the experiment show that at such shallow depths, a soliton undergoes
strong refraction and transformation. In this article, we present
hindcast simulation of a particular soliton observed during the
experiment, using a fully nonlinear, nonhydrostatic, three-dimensional
model and the actual bathymetry from the ASIAEX area. The computation
begins with the soliton at a distance about 100 km from the shelf and
obtains the propagation and evolution of the soliton over the
shelf-slope. The three-dimensional hindcast is able to reproduce the
refraction of the soliton propagation observed during the experiment as
well as the propagation speed and direction in the range observed. The
final nonlinear transformation of the soliton from symmetrical to skewed
elongated waveform is also obtained in the model consistent with the
observations. The hindcast simulation reveals that the relative position
of soliton's vorticity core to the local water depth is a crucial
indicator of the onset of transformation and formation of elevation
Full-text · Article · Jan 2009 · Journal of Geophysical Research Atmospheres
[Show abstract][Hide abstract] ABSTRACT: We focus on inverting the surface temperature (or heat) equation to obtain the surface velocity field in the coastal ocean and compare the results with those from the maximum cross correlation (MCC) technique and with the in situ velocity fields measured by the Rutgers University Coastal Ocean Dynamics Radar (CODAR). When compared with CODAR fields, velocities from the heat equation and MCC have comparable accuracies, but the heat equation technique better resolves the finer scale flow features. We use the results to directly calculate the surface divergence and vorticity. This is possible because we convert the traditionally underdetermined heat inversion problem to an overdetermined one without constraining the velocity field with divergence, vorticity, or energy statements. Because no a priori assumptions are made about the vorticity, it can be calculated directly from the velocity results. The derived vorticity field has typical open-ocean magnitudes ( ~ 5 times 10<sup>-5</sup>/s) and exhibits several structures (a warm core ring, Gulf Stream filament, and a diverging flow) consistent with the types of flows required to kinematically deform the sea surface temperature patterns into the observed configurations.
Full-text · Article · Dec 2008 · IEEE Transactions on Geoscience and Remote Sensing
[Show abstract][Hide abstract] ABSTRACT: To the extent that sea surface temperature and colors can be considered passive tracers, their motions can be tracked to estimate the current velocities, or a conservation equation can be invoked to relate their temporal variations to the velocities. We investigate the latter, the so-called tracer inversion problem, with a particular focus on (1) the conditions under which the problem can be rendered over-determined for least squares solutions, (2) the possibility of using the tracer conservation equation within the “velocity projection” framework to estimate subsurface current profiles in shallow coastal waters, and (3) the accuracy of the tracer inversion calculation in terms of the data resolution and noise. The velocity projection framework refers to relating surface motion, either measured directly or made visible by tracers, to the subsurface current motion through the equations of motion. The accuracy of the tracer inversion calculation is quantified in terms of the spatial and temporal resolution of the tracer distribution. In the presence of irreducible tracer noise, the accuracy of the inversion rapidly degrades, and it is shown that the inversion with velocity projection can help improve accuracy. The tracer inversion method developed in this study is applied to the satellite sea surface temperature data, and the velocity result is compared to the velocity measurements made with the shore-based HF Coastal Current Radar. The potential of improving the velocity estimation with the present approach is indicated.
No preview · Article · Apr 2008 · Continental Shelf Research
[Show abstract][Hide abstract] ABSTRACT: A non-hydrostatic density-stratified hydrodynamic model with a free surface has been developed from the vorticity equations rather than the usual momentum equations. This approach has enabled the model to be obtained in two different forms, weakly non-hydrostatic and fully non-hydrostatic, with the computationally efficient weakly non-hydrostatic form applicable to motions having horizontal scales greater than the local water depth. The hydrodynamic model in both its weakly and fully non-hydrostatic forms is validated numerically using exact nonlinear non-hydrostatic solutions given by the Dubriel–Jacotin–Long equation for periodic internal gravity waves, internal solitary waves, and flow over a ridge. The numerical code is developed based on a semi-Lagrangian scheme and higher order finite-difference spatial differentiation and interpolation. To demonstrate the applicability of the model to coastal ocean situations, the problem of tidal generation of internal solitary waves at a shelf-break is considered. Simulations carried out with the model obtain the evolution of solitary wave generation and propagation consistent with past results. Moreover, the weakly non-hydrostatic simulation is shown to compare favorably with the fully non-hydrostatic simulation. The capability of the present model to simulate efficiently relatively large scale non-hydrostatic motions suggests that the weakly non-hydrostatic form of the model may be suitable for application in a large-area domain while the computationally intensive fully non-hydrostatic form of the model may be used in an embedded sub-domain where higher resolution is needed.
No preview · Article · Nov 2004 · Journal of Computational Physics
[Show abstract][Hide abstract] ABSTRACT: A nonhydrostatic, hydrodynamic model of the sound speed field in a continental shelf-break environment has been developed and implemented. The model is based on a vorticity formulation of the equations of motion for an incompressible fluid with a free ocean surface, and it is capable of simulating the generation and propagation of internal tides and solibores under tidal forcing. The model has been benchmarked with an exact numerical solution for a soliton. A set of space and time evolving sound speed distributions is integrated with a parabolic equation code to compute time and frequency dependent pressure fields. Two-dimensional examples of broad-band signal gain degradation on vertical arrays in this environment are presented, as well as range-frequency maps that illustrate the structure of the waveguide invariant in a shelf-break environment that is changing in time. Implications for source localization are considered. [Work supported by ONR.]
No preview · Article · Jan 2003 · The Journal of the Acoustical Society of America
[Show abstract][Hide abstract] ABSTRACT: The ubiquitous occurrence of submesoscale cyclonic spirals in the sea as inferred from space imagery is interpreted in terms of the inertial instability of a horizontally sheared current in the oceanic mixed layer. The instability is shown to weaken anticyclonic current shear while enhancing cyclonic shear, which, in turn, becomes unstable and creates a cyclonic vortex; concurrently, surface tracer particles concentrated along the evolving cyclonic shear are wound up into a spiral, mimicking the spiral slick patterns seen in the imagery. The entire process, investigated with a fully nonlinear nonhydrostatic 3D numerical model, is contrasted with the baroclinic frontal process considered previously. The differences point to a clear need for field observations of this significant phenomenon, which are presently almost totally lacking.
Preview · Article · Dec 2002 · Geophysical Research Letters
[Show abstract][Hide abstract] ABSTRACT: An approach is described that projects surface current observations
downward to obtain subsurface current structure consistent with the
interior current dynamics. The projection approach is unrestricted by
water depth. However, this study emphasizes well-mixed constant density
flow in water depths appropriate for mid and outer shelf, where viscous
and inertial processes are comparably important. By means of twin
experiments, in which the projected current profiles are compared to the
known simulated current profiles, it is shown that both the projection
time step and the time domain affect the accuracy of the projection. At
minimum, the time domain needs to span the dominant period of current
oscillation while the time step resolves this oscillation. When both are
achieved, the projected current profiles converge to the simulated
profiles, and this convergence can be faster than that by assimilating
surface observations to correct model spin-up from an inaccurately known
initial condition. Furthermore, the projection is shown to be robust in
the presence of data noise, and with appropriate weighting of the data
constraints, the effect of noise on the projection accuracy can be
minimized. In the present projection problem, the sea surface slope is
assumed unknown and obtained together with the current profile. The use
of variable eddy viscosity in velocity projection is illustrated as well
with an iterative procedure.
Preview · Article · Nov 2002 · Journal of Geophysical Research Atmospheres
[Show abstract][Hide abstract] ABSTRACT: In situ observation and remote sensing imagery reveal the presence of longitudinal velocity convergences over bathymetric channels in tidal estuaries. We present the results of numerical experiments designed to investigate the cause of these convergences for channels possessing shallow shoal regions and a deeper central region. The equations of motion for a homogeneous fluid on a rotating Earth are solved using a fully spectral code in the across-estuary (i.e., the vertical or x-z) plane, while no alongestuary flow variations (in the y direction) are permitted. A Gaussian-shaped bottom bathymetry is chosen. In the along-channel (y) direction we impose a pressure gradient which is the sum of constant and fluctuating parts to simulate the steady and tidally oscillating parts of the estuarine flow. The details of the transient response can be complicated, but we observe that for most (~80%) of the tidal cycle there exists a cross-estuary recirculation cell collocated with a localized along-channel jet. Both of these are situated over the bottom bathymetric groove; the circulation is always clockwise when facing down current. This feature results from the generation of stream-wise vorticity through the tilting of planetary vorticity by the vertical shear of the along-estuary flow. A surface convergence-divergence pair is associated with the flow. The maximum value of each is seen to occur on the edge of the bathymetric feature but may migrate toward or away from the center as long as the current continues in the same direction. When the tide reverses, the feature reappears on the opposite shoal, and the migration of the convergence and divergence extrema begins again. We also find that the responses are qualitatively similar for all bathymetric grooves, even asymmetrically situated ones, provided that the estuary width-to-depth ratio is of order 100 or larger, the Rossby numbers are of order unity, and the Ekman layer thickness-to-channel-depth ratio is greater than ~0.65.
No preview · Article · Nov 2001 · Journal of Geophysical Research Atmospheres
[Show abstract][Hide abstract] ABSTRACT: Sea surface currents in coastal oceans are accessible to continuous direct observations by shore-based high-frequency Doppler radar systems. Inferring current structure in shallow water from such surface current observations is attempted. The approach assumes frictionally dominated flow and vertically varying current velocity on the scale of the Ekman boundary layer. The approximation of the velocity variation with depth is consequently derivable in terms of orthogonal basis functions from the sea surface kinematic and dynamic boundary conditions; specifically, the viscous momentum and shear equations evaluated at the sea surface. The inference procedure developed is demonstrated with sea surface data obtained in the coastal High-Resolution Remote Sensing Experiment on the continental shelf off Cape Hatteras. Despite uncertainties in the surface measurements, qualitative agreement is obtained between the inferred subsurface current and the current measured in situ. The sensitivity of the inference to the measurement uncertainties as well as to the model assumptions is investigated, and the inferred result is found to be generally robust.
Preview · Article · Apr 2001 · Journal of Geophysical Research Atmospheres
[Show abstract][Hide abstract] ABSTRACT: In situ observation and remote sensing imagery indicate the presence of velocity convergences located over bathymetric channels in the mouths of tidal estuaries. In this paper we present the results of numerical simulations performed to investigate these velocity structures in a rotating channel having a single bathymetric groove. The equations of motion for a homogeneous fluid on a rotating Earth are solved using a fully spectral code in the across-channel (i.e., the vertical or x-z) plane. No along-channel flow variations (in the y direction) are permitted. The bottom bathymetry is formed using a unique virtual surface approach [Goldstein et al., 1993] that generates a no-slip bottom using feedback forcing. A Gaussian-shaped channel is employed to simulate typical estuarine bathymetry. In the along-channel direction a constant pressure gradient is imposed, and the flow evolves until a steady state results. The simulations are performed at high Rossby number (of order unity) based on the width of the groove and a typical surface velocity. Simulations show the development of a localized along-channel jet colocated with an across-channel recirculation cell. This feature results from the generation of streamwise vorticity through the tilting of planetary vorticity by the vertical shear in the along-channel flow. The associated across-channel surface flow above the jet exhibits convergent and divergent regions, which correlate reasonably well with features reported previously in the literature. Their number, position, and strength are seen to vary with the along-channel Reynolds number, Ekman layer thickness, and channel aspect ratio.
No preview · Article · Apr 2000 · Journal of Geophysical Research Atmospheres
[Show abstract][Hide abstract] ABSTRACT: This work examines the stability of surface frontogenesis in the presence of a horizontal density gradient and an ageostrophic current. We pose an initial value problem, in which two homogeneous bodies of water with different densities are separated by a horizontal transition region for the density. A surface current jet flows along the density front, and the geostrophic adjustment process is simulated using a fully nonlinear pseudo spectral numerical calculation in a 10 km × 30 m range and depth domain. We allow the evolution of the surface current jet but do not permit its variation in the y direction (perpendicular to the computational domain). A number of simulations are performed for a wide range of density differences and jet strengths. Surface frontogenesis and a tendency toward geostrophic adjustment of the initially ageostrophic fields do not always exhibit a smooth subsurface circulation accompanying the bunching of the surface isopycnals. Instead, a vortex is sometimes shed from the vicinity of the evolving front, and the isopycnals are distorted by this smaller-scale vortical flow. To determine the source of the secondary symmetric baroclinic instability, the acceleration potential of the individual terms in the vorticity equation is calculated. The instability is caused by the vertical shear in the along-front jet, which is intensified by the advection and vortex-tilting processes during the frontogenesis. Although this vortex is left behind by the propagating hydraulic jump, it subsequently matures into a secondary hydraulic jump on its own. We found that in marginally unstable cases an increase in the kinematic viscosity can suppress its occurrence. Finally, we show that the unstable vortex is separate and distinct from the captured turbulent rotor which is thought to be locally trapped at a location just behind the propagating front.
No preview · Article · May 1999 · Journal of Geophysical Research Atmospheres
[Show abstract][Hide abstract] ABSTRACT: A horizontal shear flow having a Rossby number, Ro, greater than unity on a rotating plane can become unstable when its shear value is less than −f, the Coriolis frequency. In this paper, this instability is investigated for an O(10 km) submesoscale, sinusoidal shear flow in a thin homogeneous fluid layer as in an oceanic mixed layer or a shallow sea. The most unstable mode is shown by a linear analysis to occur in a narrow localized region centered around the maximum anticyclonic current shear. However, nonlinear numerical calculations show that the instability can grow to encompass both unstable and stable regions of the current. A consequence of this finite-amplitude evolution is the formation of surface convergence/shear fronts. The possibility that inertial instability mechanism is a source of some surface convergence/shear features seen in remote sensing images of the sea surface is discussed. A comparison is made with the shear-flow instability that can occur concurrently in a sinusoidal shear current, and inertial instability is shown to be the dominant instability mechanism in the immediate range above Ro=2.
No preview · Article · May 1998 · Dynamics of Atmospheres and Oceans
[Show abstract][Hide abstract] ABSTRACT: This paper deals with frontogenesis in the presence of ageostrophic vertical current shears and horizontal density gradients. The problem has broad application to the situation encountered in tidal fronts and current system meanders, but specific focus here is on Gulf Stream meander crests and filaments that advance onto the continental shelf just north of Cape Hatteras. These occur typically every few days as Gulf Stream meanders progress northeastward through the South Atlantic Bight and past Cape Hatteras. We model the submesoscale evolution of the interface between the continental shelf water and these Gulf Stream features while they are on the continental shelf. We assume the region to be characterized by an initial condition consisting of a horizontal density transition region and an ageostrophic, surface-intensified horizontal flow. The ensuing frontogenesis process is modeled numerically with an f plane calculation employing the full nonlinear equations in the depth/cross-front plane; flow is assumed out of this plane (along the front), but no variation of the flow in this direction is allowed. A pseudospectral model is employed using trigonometric functions in the horizontal and Chebyshev polynomials in the vertical. Many different scenarios are investigated by changing the width, shape, and relative positions of the density transition and velocity jet. In the majority of cases a propagating hydraulic jump is formed. Simultaneously, the initial surface jet evolves to a subsurface-intensified jet while it weakens and ultimately changes directions. The presence of this strong velocity jet can substantially enhance the rate of jump formation or completely inhibit frontogenesis. Supporting analytical calculations are used to show that the presence of vertical ageostrophic shear can augment or oppose the usual frontogenesis mechanism present when the collapsing horizontal density gradient is acted on by the resulting convergent surface current. The outcome of the shear/density gradient interaction depends upon the position of each field with respect to the other. In the vicinity of the nose of the hydraulic jump for the cases investigated, the density is seen to have a qualitatively similar dependence upon the stream function in the translating frame, irrespective of the initial condition from which it evolved.
No preview · Article · Jan 1996 · Journal of Geophysical Research Atmospheres