Rossby Radius Of Deformation Research Articles

Formation of a coastal barrier jet in the Gulf of Mexico due to the interaction of cold fronts with theSierraMadreOriental mountain range

The passage of three cold fronts (CFs) over the Gulf of Mexico was simulated with the Weather Research and Forecasting model to study the mechanisms associated with the formation of a jet parallel to the Sierra Madre Oriental (SMOr) mountain range near the port of Veracruz. Dew‐point temperature (Td= 20 °C) and available convective potential energy best simulated the horizontal propagation of the intenseCFsover the Gulf, and equivalent potential temperature helped identify the vertical position of the fronts. Additional simulations with reduced topography showed that mountain heights exceeding 2.5 km, where theSMOris closest to the coast, were crucial for the formation of a coastal barrier jet. The Froude number (Fr < 1) and the ageostrophic component of the wind at low levels were used to identify the formation of the jet; the Rossby radius of deformation determined its horizontal extent which was 81 km into the Gulf of Mexico in agreement with scatterometer winds during one of theCFs. The interaction of the northeasterly winds (known as Nortes) with the topography produced a blocking effect resulting in a stable region of cold air, a rain‐shadow and a coastal barrier jet of northwesterly direction, which generated strong cold air advection and subsidence in comparison with adjacent zones. The jet region appears as a minimum in the winter precipitation climatology indicating that the barrier jet is an important intraseasonal feature. In two of the case‐studies, maximum jet velocities reached those of a tropical storm (>18 m s−1) and occurred near the surface (<1 km above mean sea level), consistent with observations and similar to other barrier jets in the world. For the first time the jet near the port of Veracruz is documented as a coastal barrier jet. TheCFsalso produced strong Tehuano gap winds.

Solutions of quasi-geostrophic turbulence in multi-layered configurations

We consider quasi-geostrophic (QG) models in two- and three-layers that are useful in theoretical studies of planetary atmospheres and oceans. In these models, the streamfunctions are given by (1+2) partial differential systems of evolution equations. A two-layer QG model, in a simplified version, is dependent exclusively on the Rossby radius of deformation. However, the f-plane QG point vortex model contains factors such as the density, thickness of each layer, the Coriolis parameter, and the constant of gravitational acceleration, and this two-layered model admits a lesser number of Lie point symmetries, as compared to the simplified model. Finally, we study a three-layer oceanography QG model of special interest, which includes asymmetric wind curl forcing or Ekman pumping, that drives double-gyre ocean circulation. In three-layers, we obtain solutions pertaining to the wind-driven doublegyre ocean flow for a range of physically relevant features, such as lateral friction and the analogue parameters of the f-plane QG model. Zero-order invariants are used to reduce the partial differential systems to ordinary differential systems. We determine conservation laws for these QG systems via multiplier methods.

Changing duration and spatial extent of midlatitude precipitation extremes across different climates

AbstractPrecipitation extremes intensify with climate warming in observations and simulations, but changes in their duration or spatial extent are not well understood. Here the duration and zonal length of midlatitude precipitation extremes are quantified in climate model simulations. Most comprehensive climate models project a decrease in duration over the 21st century, although the magnitude of the decrease with warming is less than 1% K−1 in the multimodel mean. An advective time scale based on the mean zonal wind is shown to be linked to the duration in terms of spatial distribution, intermodel differences, and response to climate change. In simulations with an idealized climate model, a stronger meridional temperature gradient decreases the duration despite increases in the zonal length, and this is explained using the thermal wind relation and the Rossby deformation radius. However, the response of the zonal length to increasing mean temperature requires further study.

Turbulent kinetic energy of the ocean winds over the Kuroshio Extension from QuikSCAT winds (1999–2009)

AbstractWe investigate mesoscale turbulence (10–1000 km) in the ocean winds over the Kuroshio Extension ( N– N, E– E) using the QuikSCAT data set (November 1999 to October 2009). We calculate the second (Djj) and third‐order structure functions (Djjj) and the spatial variance (Vj) as a function of scale r ( denotes, respectively, the longitudinal (divergent) and transverse (vortical) component). The most interesting results of the analysis follow. Although both and measure the turbulent kinetic energy (TKE), we find that is the more robust measure. The spatial variance density (dVj/dr) has a broad peak near 450 km (close to the midlatitude Rossby radius of deformation). On interannual time scales, TKE correlates well with the El Niño 3.4 index. According to turbulence theory, the kinetic energy cascades downscale (upscale) if (also skewness ) is negative (positive). Our results for the Kuroshio Extension are consistent with a downscale cascade (indicating convergence dominates). Furthermore, classical turbulence theory predicts that and independent of r; however, we find SL varies strongly with r, from −4 at small scales to −0.3 at large scales. This nonclassical behavior implies strong‐scale interaction, which we attribute to the rapid, and sometimes explosive, growth of storms in the region through baroclinic instability. Finally, we find that ST (a measure of cyclonic/anticyclonic asymmetry) is positive (cyclonic) and also varies strongly with r, from 4 at small scales to 0.5 at large scales. New turbulence models are needed to explain these results, and that will benefit Weather Prediction and climate modeling.

Eddy properties in the Subtropical Countercurrent, Western Philippine Sea

An array of six oceanographic moorings with acoustic and environmental sensors was deployed in the central Philippine Sea from April 2010 to April 2011. The location spanned 18−23°N, 124 – 130°E in the Subtropical Countercurrent (STCC). The most prominent feature of the data set was a densely-packed eddy field with about equal numbers of cyclones and anticyclones moving westward at 6–12kmd−1. Eddies of either sign displaced the thermocline about ±50m and had surface velocities exceeding 110cms−1. While warm eddies were slightly larger than cold eddies, the distance to maximum radial velocity was similar for both at about 65km, close to the local Rossby radius of deformation. The steepness parameter U/c in the eddies ranged from 3 to 10, accompanied by relative vorticity of order 0.1–0.3f, suggesting nonlinear, quasigeostrophic features with trapped cores rather than linear waves. This was borne out by the water mass analysis which showed high salinity, high spice North Pacific Tropical Water (NPTW) being transported westward in the warm eddy cores. The total KE and APE in eddies of both signs was about 1×1015J with 85% of the APE and 74% of the KE located above 250m depth. This equipartitioning of energy suggests mature eddies near equilibrium, that had been evolving for some time as they propagated into the area from the east.

Convectively Induced Stabilizations and Subsequent Recovery with Supercell Thunderstorms during the Mesoscale Predictability Experiment (MPEX)

Abstract The adiabatic and diabatic processes inherent to midlatitude deep convective storms are well known to modify the atmospheric temperature, moisture, and winds especially within horizontal scales equivalent to a Rossby radius of deformation. Such modifications, or “feedbacks,” induced by supercell thunderstorms were a particular focus of the Mesoscale Predictability Experiment (MPEX), owing to the unique supercell dynamics and associated supercell intensity and longevity. During the MPEX field phase, which was conducted 15 May–15 June 2013 within the Great Plains region of the United States, radiosonde observations collected in immediate supercell wakes exhibited temperature lapse rates that were qualitatively and quantitatively similar to preconvective lapse rates above the boundary layer. Complementary idealized model simulations were used to confirm that there was little residual effect of the supercell in the wake of the moving storm except within the area occupied by the surface cold pool, and where stabilizations were induced adiabatically by transient gravity wave disturbances. The persistency of the (i) cold pool, and its inhibition to surface-based convection, depended on the evolving cold pool strength and environmental winds; and (ii) gravity wave effects depended on the Doppler-shifted phase speed relative to the moving storm. Otherwise, recovery of the wake environment to its preconvective state occurred approximately over a time scale defined by the updraft length scale and horizontal advective velocity scale.

The impact of wave number selection and spin‐up time in spectral nudging

Numerous studies have been presented describing the benefits of using Nudging to constrain the evolution of a numerical model. Recently, a variation of this approach called Spectral Nudging has gained popularity for its ability to maintain the higher temporal and spatial variability of the model results, while forcing the large scales in the solution with a coarser‐resolution field. Here, we focus on a not‐much‐explored aspect of this technique: the impact of selecting different cut‐off wave numbers and spin‐up times. We perform 4‐day‐long simulations with the WRF model, daily for three different 1‐month periods including a free run and several Spectral Nudging experiments with cut‐off wave numbers ranging from the smallest to the largest possible (Grid Nudging). Results show that Spectral Nudging is very effective at imposing the selected synoptic scales onto the solution, while allowing the limited‐area model to incorporate finer‐scale features. The model error diminishes rapidly as the nudging expands over broader parts of the spectrum, but this decreasing trend ceases sharply at cut‐off wave numbers equivalent to a length‐scale of about 1000 km, and the error magnitude changes minimally thereafter. This scale corresponds to the Rossby radius of deformation, separating synoptic from convective scales in the flow. When nudging is applied for scales larger than the Rossby radius of deformation, a shifting of the synoptic patterns can occur in the solution, yielding large model corrections towards the analysis. However, when selecting smaller scales, the fine‐scale contribution of the model is damped, thus making 1000 km the appropriate scale threshold to nudge in order to balance both effects. Finally, we note that longer spin‐up times are needed for model errors to stabilize when using Spectral Nudging than with Grid Nudging. Results suggest that this time is between 36 and 48 h.

The Arithmetic Geometry of Resonant Rossby Wave Triads

Linear wave solutions to the Charney--Hasegawa--Mima equation with periodic boundary conditions have two physical interpretations: Rossby (atmospheric) waves, and drift (plasma) waves in a tokamak. These waves display resonance in triads. In the case of infinite Rossby deformation radius, the set of resonant triads may be described as the set of integer solutions to a particular homogeneous Diophantine equation, or as the set of rational points on a projective surface $X$. The set of all resonant triads was found by Bustamante and Hayat (2013) via mapping to quadratic forms. Our work independently finds all resonant triads via a rational parametrization of $X$. We provide a fiberwise description of $X$ as a rational singular elliptic surface, yielding many new results about the set of wavevectors belonging to resonant triads. In particular, we show there is an infinite number of resonant triads (with relatively prime wavevectors) containing a wavevector $(a,b)$ with $a/b=r$, where $r$ is any given rationa...

Assessing Upper Tropospheric Jet Streak Proximity Using the Rossby Radius of Deformation

The Rossby radius of deformation is a parameter that describes the relative role of buoyant and inertial forces for atmospheric phenomena in a flow regime. It will be demonstrated that it can also be used to determine whether or not forcing for vertical motions in the region between upper level tropospheric jet streaks overlaps or interacts. Using predefined points in the entrance and exit regions of neighboring upper level jet streaks, the distance between them is calculated for each event. If they are closer than twice the Rossby radius of deformation, the resulting region affected by both streaks is termed the Rossby Radius of Deformation Overlap Zone (RRDOZ). Plan-view and cross-sectional analysis shows that ageostrophic transverse circulations within the RRDOZ led to enhanced upward vertical velocities as predicted in prior research. Lastly, a short-term climatology for overlap events in North America is derived, and these are classified according to three proposed archetypes.

Improving Mesoscale Altimetric Data From a Multitracer Convolutional Processing of Standard Satellite-Derived Products

Multisatellite measurements of altimeter-derived sea surface height (SSH) have provided a wealth of information on the ocean. Yet, horizontal scales below 100 km remain scarcely resolved. Especially, in the Mediterranean Sea, an important fraction of the mesoscale range, characterized by a small Rossby radius of deformation of 15–20 km, is not properly retrieved by altimeter-derived gridded products. Here, we investigate a novel processing of AVISO products with a view to resolving the horizontal scales sensed by current along-track altimeter data. The key feature of our framework is the use of linear convolutional operators to model the fine-scale SSH detail as a function of different sea surface fields, especially optimally interpolated SSH and sea surface temperature (SST). The proposed model embeds the surface quasi-geostrophic SST–SSH synergy as a special case. Using an observing system simulation experiment with simulated SSH data from model outputs in the Western Mediterranean Sea, we show that the proposed approach has the potential for improving current optimal interpolations of gridded altimeter-derived SSH fields by more than 20% in terms of relative SSH and kinetic energy mean square error, as well as in terms of spectral signatures for horizontal scales ranging from 30 to 100 km. Our results also suggest that SST–SSH relationship may only play a secondary role compared with the interscale SSH cascade. We further discuss the relevance of the proposed approach in the context of future altimetric satellite missions.

A regime diagram for ocean geostrophic turbulence

A two‐dimensional regime diagram for geostrophic turbulence in the ocean is constructed by plotting observation‐based estimates of the non‐dimensional eddy length‐scale against a nonlinearity parameter equal to the ratio of the root‐mean‐square eddy velocity and baroclinic Rossby phase speed. Two estimates of the eddy length‐scale are compared: the equivalent eddy radius inferred from the area enclosed by contours of sea‐surface height, and the ‘unsuppressed’ mixing length, based on an estimate of the eddy diffusivity with mean flow effects removed. For weak nonlinearity, as found in the Tropics, the mixing length mostly corresponds to the stability threshold for baroclinic instability whereas the eddy radius corresponds to the Rhines scale; it is suggested that this mismatch is indicative of the inverse energy cascade that occurs at low latitudes in the ocean and the zonal elongation of eddies. At larger values of nonlinearity, as found at mid‐ and high latitudes, the eddy length‐scales are much shorter than the stability threshold, within a factor of 2.5 of the Rossby deformation radius.

The Effect of Eddy–Eddy Interactions on Jet Formation and Macroturbulent Scales

Abstract The effect of eddy–eddy interactions on zonal and meridional macroturbulent scales is investigated over a wide range of eddy scales, using high-resolution idealized GCM simulations with and without eddy–eddy interactions. The wide range of eddy scales is achieved through systematic variation of the planetary rotation rate and thus multiple-jet planets. It is found that not only are eddy–eddy interactions not essential for the formation of jets, but the existence of eddy–eddy interactions decreases the number of eddy-driven jets in the atmosphere. The eddy–eddy interactions have little effect on the jet scale, which in both types of simulations coincides with the Rhines scale through all latitudes. The decrease in the number of jets in the presence of eddy–eddy interactions occurs because of the narrowing of the latitudinal region where zonal jets appear. This narrowing occurs because eddy–eddy interactions are mostly important at latitudes poleward of where the Rhines scale is equal to the Rossby deformation radius. Thus, once eddy–eddy interactions are removed, the conversion from baroclinic to barotropic eddy kinetic energy increases, and eddy–mean flow interactions intrude into these latitudes and maintain additional jets there. The eddy–eddy interactions are found to increase the energy-containing zonal scale so it coincides with the jets’ scale and thus make the flow more isotropic. While the conversion scale coincides with the most unstable scale, the Rossby deformation radius does not provide a good indication to these scales in both types of simulations.

Rossby wave Green’s functions in an azimuthal wind

Green’s functions for Rossby waves in an azimuthal wind are obtained, in which the stream-function depends on r, and t, where r is cylindrical radius and is the azimuthal angle in the -plane relative to the easterly direction, in which the x-axis points east and the y-axis points north. The Rossby wave Green’s function with no wind is obtained using Fourier transform methods, and is related to the previously known Green’s function obtained for this case, which has a different but equivalent form to the Green’s function obtained in the present paper. We emphasise the role of the wave eikonal solution, which plays an important role in the form of the solution. The corresponding Green’s function for a rotating wind with azimuthal wind velocity (const.) is also obtained by Fourier methods, in which the advective rotation operator in position space is transformed to a rotation operator in transform space. The finite Rossby deformation radius is included in the analysis. The physical characteristics of the Green’s functions are delineated and applications are discussed. In the limit as , the rotating wind Green’s function reduces to the Rossby wave Green function with no wind.

The latitudinal dependence of the oceanic barotropic eddy kinetic energy and macroturbulence energy transport

AbstractGeostrophic turbulence theory predicted already a few decades ago an inverse energy cascade in the barotropic mode, yet there has been limited evidence for it in the ocean. In this study, the latitudinal behavior of the oceanic barotropic energy balance and macroturbulent scales is studied using the ECCO2 (Estimating the Circulation and Climate of the Ocean) state estimate, which synthesizes satellite data and in situ measurements with a high‐resolution general circulation model containing realistic bathymetry and wind forcing. It is found that inverse energy cascade occurs at high latitudes, as eddy‐eddy interactions spread the conversion of eddy kinetic energy from the baroclinic to the barotropic mode, both upscale and downscale. At these latitudes, the conversion scale of baroclinic eddy kinetic energy and the energy‐containing scale follow the most unstable and Rhines scales, respectively. Even though an inverse energy cascade occurs at high latitudes, the energy spectrum follows a steeper slope than the −5/3 slope. Different than classic arguments, the Rossby deformation radius does not follow the baroclinic conversion and most unstable scales.

BAROCLINIC INSTABILITY IN THE SOLAR TACHOCLINE. II. THE EADY PROBLEM

ABSTRACT We solve the nongeostrophic baroclinic instability problem for the tachocline for a continuous model with a constant vertical rotation gradient (the Eady problem), using power series generated by the Frobenius method. The results confirm and greatly extend those from a previous two-layer model. For effective gravity G independent of height, growth rates and ranges of unstable longitudinal wavenumbers m and latitudes increase with decreasing G. As with the two-layer model, the overshoot tachocline is much more unstable than the radiative tachocline. The e-folding growth times range from as short as 10 days to as long as several years, depending on latitude, G, and wavenumber. For a more realistic effective gravity that decreases linearly from the radiative interior to near zero at the top of the tachocline, we find that only m = 1, 2 modes are unstable, with growth rates somewhat larger than for constant G, with the same value as at the bottom of the tachocline. All results are the same whether we assume that the vertical velocity or the perturbation pressure is zero at the top of the layer; this is a direct consquence of not employing the geostrophic assumption for perturbations. We explain most of the properties of the instability in terms of the Rossby deformation radius. We discuss further improvements in the realism of the model, particularly adding toroidal fields that vary in height, and including latitudinal gradients of both rotation and toroidal fields.

Modeling the Interaction between Quasigeostrophic Vertical Motion and Convection in a Single Column

Abstract A single-column modeling approach is proposed to study the interaction between convection and large-scale dynamics using the quasigeostrophic (QG) framework. This approach extends the notion of “parameterization of large-scale dynamics,” previously applied in the tropics via the weak temperature gradient approximation and other comparable methods, to the extratropics, where balanced adiabatic dynamics plays a larger role in inducing large-scale vertical motion. The diabatic heating in an air column is resolved numerically by a single-column model or a cloud-resolving model. The large-scale vertical velocity, which controls vertical advection of temperature and moisture, is computed through the QG omega equation including the dry adiabatic terms and the diabatic heating term. The component due to diabatic heating can be thought of as geostrophic adjustment to that heating and couples the convection to the large-scale vertical motion. The approach is demonstrated using two representations of convection: a single-column model and linear response functions derived by Z. Kuang from a large set of cloud-resolving simulations. The results are qualitatively similar in both cases. The behavior of convection that is strongly coupled to large-scale dynamics is significantly different from that in the uncoupled case. The positive feedback of the diabatic heating on the large-scale vertical motion reduces the stability of the system, extends the decay time scale after initial perturbations, and increases the amplitude of convective responses to transient large-scale perturbations or imposed forcings. The diabatic feedback of convection on vertical motion is strongest for horizontal wavelengths on the order of the Rossby deformation radius.

Interference pattern of the sound field in the presence of an internal Kelvin wave in a stratified lake.

Internal Kelvin waves (IKWs) initiated by rotation of the Earth are one of the main hydrodynamic phenomena in large stratified lakes where baroclinic Rossby radius of deformation is smaller than the horizontal scale of the lake. IKWs can be identified using the spectra of internal waves, where in the presence of IKWs, the inertial frequency is at maximum. IKWs play a rather important role in the lake's dynamics for different processes, both in the water layer and sediment, especially at the periphery of lake. Due to influence of internal waves on the sound propagation, acoustical methods can be used for estimation of behaviour of IKWs. In this paper, the spatiotemporal variability of the mid-frequency (∼1 kHz) sound field in the presence of IKWs in a deep stratified Lake Kinneret is studied using numerical simulations based on normal-mode theory. Due to the specific character of perturbation of the water layer, IKWs can cause specific variations of interference pattern, in particular, a significant shift of the sound interference pattern both in spatial and frequency domain. These shifts can be easily measured and used for reconstruction of IKW parameters.

Circulation in the Arctic Ocean: Results from a high-resolution coupled ice-sea nested Global-FVCOM and Arctic-FVCOM system

A high-resolution unstructured-grid global-regional nested ice-current coupled FVCOM system was configured for the Arctic Ocean and used to examine the impact of model resolution and geometrical fitting on the basin-coastal scale circulation and transport in the pan-Arctic. With resolving steep bottom slope and irregular coastal geometry, the model was capable of simulating the multi-scale circulation and its spatial variability in the Arctic Basin and flow through the Bering Strait, Fram Strait and Canadian Archipelago. The model-simulated annual-mean velocities were in good agreement with observations within the measurement uncertainty and variability due to insufficient sampling. The errors in the flow direction varied with the flow speed, larger in the weak velocity zone and smaller as the velocity increased. In the upper 50-m layer, the annual-mean circulation pattern was dominated by the wind- and ice-drifting-induced anticyclonic circulation in the Arctic Basin and a relatively strong cyclonic slope current along the edge of the continental shelf. In the deep 200–600-m layer, a relatively permanent cyclonic circulation occurred along the steep bottom slope. These annual-mean circulations accounted for ∼85% of the total kinetic energy variance. De-trending the mean flow, an empirical orthogonal function (EOF) analysis showed that the semi-annual and seasonal variability of the sub-tidal flow was dominated by the first and second modes that accounted for ∼46% and ∼30% of the total variance in the upper 50-m layer and ∼58% and 20% in the deep 200–600-m layer. Consistent with observations, the AO-FVCOM-simulated cyclonic slope flow was characterized by a large positive topostrophy. Sensitivity experiment results with various grid configurations suggested that the currents over slopes, narrow straits and water passages featured topographic and baroclinic frontal dynamical scales associated with bathymetric slope and internal Rossby deformation radius. Over the Arctic slope, since these two scales are in the same order, the along-slope current could be captured, as the cross-isobath model resolution was refined to resolve the steep bottom topography. Under this condition, there is no need to add Neptune forcing into the momentum equations. The accuracy of the estimation of the transport through the strait and narrow water passage was affected by the model resolution. In Fram Strait where the flow is characterized by strong lateral current shear resulting from the Atlantic inflow and Arctic outflow, the transport estimation could have a significant uncertainty due to both horizontal and vertical sampling resolutions.

Inertia-gravity wave radiation from the merging of two co-rotating vortices in the f-plane shallow water system

Inertia-gravity wave radiation from the merging of two co-rotating vortices is investigated numerically in a rotating shallow water system in order to focus on cyclone–anticyclone asymmetry at different values of the Rossby number (Ro). A numerical study is conducted on a model using a spectral method in an unbounded domain to estimate the gravity wave flux with high accuracy. Continuous gravity wave radiation is observed in three stages of vortical flows: co-rotating of the vortices, merging of the vortices, and unsteady motion of the merged vortex. A cyclone–anticyclone asymmetry appears at all stages at smaller Ro (≤20). Gravity waves from anticyclones are always larger than those from cyclones and have a local maximum at smaller Ro (∼2) compared with that for an idealized case of a co-rotating vortex pair with a constant rotation rate. The source originating in the Coriolis acceleration has a key role in cyclone–anticyclone asymmetry in gravity waves. An additional important factor is that at later stages, the merged axisymmetric anticyclone rotates faster than the elliptical cyclone due to the effect of the Rossby deformation radius, since a rotation rate higher than the inertial cutoff frequency is required to radiate gravity waves.

A Conservative Formulation and a Numerical Algorithm for the Double-Gyre Nonlinear Shallow-Water Model

AbstractWe present a conservative formulation and a numerical algorithm for the reduced-gravity shallow-water equations on a beta plane, subjected to a constant wind forcing that leads to the formation of double-gyre circulation in a closed ocean basin. The novelty of the paper is that we reformulate the governing equations into a nonlinear hyperbolic conservation law plus source terms. A second-order fractional-step algorithm is used to solve the reformulated equations. In the first step of the fractional-step algorithm, we solve the homogeneous hyperbolic shallow-water equations by the wave-propagation finite volume method. The resulting intermediate solution is then used as the initial condition for the initial-boundary value problem in the second step. As a result, the proposed method is not sensitive to the choice of viscosity and gives high-resolution results for coarse grids, as long as the Rossby deformation radius is resolved. We discuss the boundary conditions in each step, when no-slip boundary conditions are imposed to the problem. We validate the algorithm by a periodic flow on an f-plane with exact solutions. The order-of-accuracy for the proposed algorithm is tested numerically. We illustrate a quasi-steady-state solution of the double-gyre model via the height anomaly and the contour of stream function for the formation of double-gyre circulation in a closed basin. Our calculations are highly consistent with the results reported in the literature. Finally, we present an application, in which the double-gyre model is coupled with the advection equation for modeling transport of a pollutant in a closed ocean basin.