Comments on the “shear effect” and diffusion in the Lagrangian framework

We indentify three different types of Lagrangian coordinate systems that are used in oceanography. These are: true Lagrangian coordinates (TLC), Lagrangian coordinates (LC), and averaged Lagrangian coordinates (ALC). The diffusion process is studied in each of these coordinate systems. At large scales the eddydiffusivity is proven to be independent of molecular diffusivity, providing the spectrum of turbulent kinetic energy varies as scale raised to a power less than 4 1/3. The shear effect is examined using solutions to the averaged Lagrangian diffusion equation obtained by Okuboet al. (1983). In Eulerian coordinates both advection and diffusion are necessary for the occurrence of the shear effect, while in ALC timedependent dispersion coefficients are necessary for the process. In TLC we use the method of Taylor (1921) to study the dispersion of material by a velocity field, that from the Eulerian perspective, consists of turbulent motion across a uniform shear. The transformation of the above Eulerian velocity field into TLC results in a uniform deformation field and turbulent motion both along and across the shear. This work shows how dispersion of material is related to the turbulent Eulerian velocity and uniform velocity gradients. The instantaneous rate of change of variance of a spreading patch of material is completely specified by the instantaneous divergence obtained over the area occupied by the patch (Kawai, 1976). This relationship is shown to depend upon the fact that at any particular instant it is possible to define TLC that are equivalent to the Eulerian coordinates. In order to describe patch spreading from divergence measured over longer periods it is also necessary to consider other dispersive processes.Contribution number of the Newfoundland Institute of Cold Ocean Science.     

Physical behaviors of the iron-fertilized patch in SEEDS II

Sulfur hexafluoride (SF₆) tracer release experiments were carried out to trace the iron-fertilized water mass during the iron-fertilization experiments in the western North Pacific of Subarctic Pacific Iron Experiment for Ecosystem Dynamics Study II (SEEDS II) in 2004. A solution of Fe and SF₆ tracer was released into the surface mixed layer over an 8×8 km area, and the fertilized patch was traced by onboard SF₆ analysis for 12 days during each experiment. A Lagrangian frame of reference was maintained by the use of a drogued GPS buoy released at the center of the patch to reduce the advection effect on observations. The patch moved along the contour of sea-surface height (SSH) of a clockwise mesoscale eddy for 4 days after release. Then strong easterly winds dragged the patch across the contour of SSH. The patch behavior was affected by both the mesoscale eddy and surface winds. Apparent horizontal diffusivities were determined by the change of the distribution of SF₆ concentrations. The averaged apparent horizontal diffusivity was about 49 m² s⁻¹ during SEEDS II. It was larger than the one in SEEDS. Mixed-layer depth (MLD) was 8.5–18 m during SEEDS, and 12–33 m during SEEDS II. The larger horizontal diffusivity and deeper MLD in SEEDS II were disadvantages to maintain a high iron concentration in the surface layer compared to SEEDS. Temporal change of the MLD corresponded to the temporal change of chlorophyll-a concentration. Temporal change in the surface MLD was also important for the response of phytoplankton by iron fertilization.     

Jellyfish patch formation investigated by aerial photography and drifter experiment

Jellyfish patch formation is investigated by conducting a drifter experiment combined with aerial photography of a sustained patch of the moon jellyfish in Hokezu Bay, Japan. Jellyfish patches are aggregations of individuals that are caused by a combination of swimming (active influence) and advection by currents (passive influence). The drifter experiment involved the injection of 49 drifters around a distinct surface patch of jellyfish within an area of approximately 300 m × 300 m. The drifters’ motion, caused only by the passive influence, was recorded in a series of 38 aerial photographs taken over approximately 1 h. The ambient uniform current field larger than the patch scale was estimated from the movement of the centroid position of drifters, while the distribution of horizontal divergence and relative vorticity around the patch was estimated from the time-derivative in areas of triangles formed by the drifters. The centroid positions of both drifters and patches moved stably toward the bay head at different speeds. The difference vector between the patch and drifter centroids was directed to the sun, and was opposite to the ambient current. The distributions of vorticity and divergence around patches exhibited inhomogeneity within the patch scale, and the drifters in this nonuniform current field aggregated near the convergence area within 1 h. The results suggest that horizontal patch formation is predominantly influenced by passive factors at the surface of Hokezu Bay. Furthermore, the upward swimming against downwelling may make sustained patch in surface layer.     

Processes governing the carbon chemistry during the SAGE experiment

Measurements of pCO₂, pH and alkalinity in the surface waters of an iron fertilised patch of sub-Antarctic water were made during SAGE (SOLAS SAGE: Surface-Ocean Lower Atmosphere Studies Air-Sea Gas Experiment). The iron addition induced a minor phytoplankton bloom, however the patch dynamics were dominated by physical processes which suppressed and masked the biological effects. The Lagrangian nature of the experiment allowed the carbonate chemistry in the patch to be followed for 15.5 days, and the relative importance of the biological and physical factors influencing the surface water pCO₂ was estimated. The pCO₂ of the surface waters of the patch increased from 327 ??atm prior to iron addition to 338 ??atm on Day 14, effects of vertical and horizontal mixing offset the 15 ??atm drawdown that would have occurred had the induced biological uptake been the sole factor to influence the pCO₂. The air-sea carbon flux calculated using the measured skin temperature and a piston velocity parameterisation determined during SAGE (Ho et al., 2006) was 98.5% of the flux determined using conventional bulk temperature measurement and the Wanninkhof (1992) piston velocity parameterisation. The skin temperature alone contributed to an 8% increase in the flux compared with that determined using bulk temperature.     

A proposed adiabatic formulation of 3‐dimensional global atmospheric models based on potential vorticity

A 2‐time‐level finite difference atmospheric general circulation model based on the semi‐Lagrangian advection of pseudo potential vorticity (which becomes potential vorticity in that part of the domain where the hybrid vertical coordinate becomes isentropic) has been formulated. At low levels, the hybrid vertical coordinate is terrain following. The problem of isentropic potential vorticity possibly becoming ill‐defined in the regions of planetary boundary layer is thus circumvented. The divergence equation is a companion to the (pseudo) potential vorticity equation and the model is thus called a PV‐D model. Many features of a previously developed shallow water PV‐D model are carried over: a modification of the PV equation needed to give computational stability of long Rossby waves; a semi‐Lagrangian semi‐implicit treatment of both the linear and the nonlinear terms; the use of an unstaggered grid in the horizontal; the use of a nonlinear multigrid technique to solve the nonlinear implicit equations. A linear numerical stability analysis of the model's gravity–inertia waves indicates that the potential temperature needs to be separated into horizontal mean and perturbation parts. This allows an implicit treatment of the vertical advection associated with the mean in the thermodynamic equation. Numerical experiments with developing baroclinic waves have been carried out and give realistic results.     

Physical behavior of the SEEDS iron-fertilized patch by sulphur hexafluoride tracer release

The first iron (Fe) – fertilization experiment in the western North Pacific was carried out using SF₆ to trace the Fe-fertilized water mass. A solution in 10,800 liters of seawater of 350 kg of Fe and 0.48 M of SF₆ tracer was released into the mixed layer over a 8 × 10 km area. On the first underway transects through the patch after the Fe release, we observed a significant increase of dissolved Fe (ave. 2.89 nM). The fertilized patch was traced for 14 days by on-board SF₆ analysis. A Lagrangian frame of reference was maintained by the use of a drogued GPS buoy released at the center of the patch. The patch moved westward at a rate of 6.8 km d⁻¹. Mixed layer depth increased from 8.5 to 15 m during the experiment. Horizontal diffusivity was determined by the change of SF₆ concentration in the patch. The horizontal diffusivity increased during the experiment. We evaluate here the fate of Fe in a Fe-fertilized patch using the dilution rate determined from sulphur hexafluoride (SF₆) concentration. Dissolved Fe concentrations subsequently decreased rapidly to 0.15 nM on Day 13. However, the dissolved Fe half-life of 43 h was relatively longer than in previous Fe-enrichment studies, and we observed a larger increase of the centric diatom standing stock and corresponding drawdown of macro-nutrients and carbon dioxide than in the previous studies. The most important reason for the larger response was the phytoplankton species in the western North Pacific. In addition, the smaller diffusivity and shallower mixed layer were effective to sustain the higher dissolved Fe concentration compared to previous experiments. This might be one reason for the larger response of diatoms in SEEDS.     

On the fundamental dynamics of barotropic circulation in shallow seas

It is suggested that the shallow sea circulation should be related to the Lagrangian residual circulation but not to the conventional Eulerjan mean circulation, and further, the first order Lagrangian residual circulation-the mass-transport velocity field is used to define the steady circulation as the lowest order shallow sea circulation.The set of equation governing the shallow sea circulation is reformulated, which shows that there are no the so-called “tidal surface source” and the “tidal stress”. A vorticity equation for the stream function of horizontal transports is derived, which is easily solved by the conventional numerical methods.An intertidal convection-diffusion equation governing the concentration of conservative and passive tracer is derived. Differing from the conventional equation, the equation derived in the present paper reveals the Lagrangian convection and has no need to introduce the artificial hypothesis for the so-called “tidal dispersion”.The theory presented here is base     

Hybrid <Emphasis Type="Italic">N</Emphasis>-order Lagrangian interpolation Eulerian-Lagrangian method for salinity calculation

The Eulerian-Lagrangian method (ELM) has been used by many ocean models as the solution of the advection equation, but the numerical error caused by interpolation imposes restriction on its accuracy. In the present study, hybrid N-order Lagrangian interpolation ELM (LiELM) is put forward in which the N-order Lagrangian interpolation is used at first, then the lower order Lagrangian interpolation is applied in the points where the interpolation results are abnormally higher or lower. The calculation results of a step-shaped salinity advection model are analyzed, which show that higher order (N=3-8) LiELM can reduce the mean numerical error of salinity calculation, but the numerical oscillation error is still significant. Even number order LiELM makes larger numerical oscillation error than its adjacent odd number order LiELM. Hybrid N-order LiELM can remove numerical oscillation, and it significantly reduces the mean numerical error when N is even and the current is in fixed direction, while it makes less effect on mean numerical error when N is odd or the current direction changes periodically. Hybrid odd number order LiELM makes less mean numerical error than its adjacent even number order LiELM when the current is in the fixed direction, while the mean numerical error decreases as N increases when the current direction changes periodically, so odd number of N may be better for application. Among various types of Hybrid N-order LiELM, the scheme reducing N-order directly to 1st-order may be the optimal for synthetic selection of accuracy and computational efficiency.     

An analytical calculation of two-dimensional dispersion

The methods of Okuboet al. (1976a) are used to calculate the Lagrangian deformations and diffusivities of a cluster of drifters. A solution of the two-dimensional first-order advection-diffusion equation (Okuboet al., 1983a) is then used to calculate the dimensions and orientation of the cluster from these Lagrangian deformations and diffusivities. The solution is shown to be internally consistent (to give cluster areas that are consistent with the observed cluster areas) to within a 0.5% error. As time progresses a larger portion of the dispersion is caused by the diffusivities rather than the deformations. In the experiments analyzed the Lagrangian deformations and diffusivities are generally observed to increase at a constant rate over time intervals of about one hour. Dimensional arguments suggest that Lagrangian diffusivities increase proportional tot ² and the deformations proportional tot ^1,5 for time intervals large compared to the period required to spread from a point source to the initial cluster dimensions. Small quadratic velocity gradients cause the solution of the first order advection-diffusion equation to overestimate cluster spreading. Most of the displacement (once motion due to the mean velocity and linear deformations is extracted) is caused by scales of motion much smaller than the cluster. This explains the relatively small magnitude of the errors caused by parameterizing quadratic and other statistically significant nonlinear shears as a component of the eddy-diffusivity.     

Wave-current force on horizontal cylinder

For the calculation of wave-current force on horizontal cylinder a modified Morison's equation is used. A redefined Keulegan- Carpenter number KC2 is determined for the horizontal cylinder in wave-current co-existing field. The force coefficients are well related to the redefined KC2 number. As to the comparison with the force on vertical cylinder, the characteristics of force on horizontal cylinder are quite similar to those on vertical cylinder, but the force coefficients for horizontal cylinder are larger than those for vertical cylinder. It is proved by the authors' calculation that the results of monochromatic wave can be used directly for the determination of irregular wave-current force on horizontal cylinder in time domain.     

Numerical and simplified methods for the calculation of the total horizontal wave force on a perforated caisson with a top cover

The VOF method and the k–ε model, combined with the equation of state of air at constant temperature, have been used to calculate the total horizontal wave force caused by monochromatic waves acting on a perforated caisson with a top cover. From comparison of various parameters, such as the total horizontal force, the pressure difference on the front wall, the pressure on the back wall and the pressure on the top cover, between the numerical results and test data, it can be seen that the numerical results agree well with the test data. It is concluded that the method described in this paper can be utilized to calculate wave forces acting on perforated caissons with a top cover in the case of nonovertopping, nonbreaking waves. A simplified method to calculate the total horizontal force has been developed, based on test data, using a least-squares method. A comparison between the numerical results and the values calculated from the simplified equations shows good agreement. Therefore the simplified equations can be used in engineering applications to evaluate the total horizontal force on a perforated caisson with a top cover.     

Composite Analysis of North Pacific Subtropical Mode Water Properties with Respect to the Strength of the Wintertime East Asian Monsoon

The interannual variation of the thermal structure of North Pacific subtropical mode water (NPSTMW) is investigated by means of composite analysis with respect to the wintertime Monsoon Index (MOI) which can represent the strength of the wintertime East Asian monsoon. The wind stress field over the NPSTMW formation area has significant variation over the interannual (2–4 year) and the decadal (10–20 year) bands. Changes in interannual variation are well correlated with the intensity of the wintertime East Asian monsoon. By means of composite analysis, it is found that significant differences occur in the thermal structure of the NPSTMW between stronger and weaker monsoon years. That is, colder and thicker NPSTMW is formed in years with stronger monsoons. Analysis of the heat flux through the sea surface and horizontal heat divergence in the Ekman layer shows that colder and thicker NPSTMW in stronger monsoon years can be attributed to a larger amount of heat release through the sea surface in the formation area. A larger horizontal divergence of the heat transport in the upper Ekman layer is considerably responsible for this increased heat loss.     

Matching carbon pools and fluxes for the Southern Ocean Iron Release Experiment (SOIREE)

The Lagrangian Southern Ocean Iron Release Experiment (SOIREE) allowed study of a gradually evolving iron-mediated phytoplankton bloom in water labelled with the inert tracer sulfur hexafluoride, SF₆. This article describes a pelagic carbon budget for the mixed layer in SOIREE and assesses the extent to which closure of the budget is achieved. Net community production (NCP) converted 837 mmol m⁻² of inorganic carbon to organic carbon in 12.0 d after the first iron addition. A large fraction (41%) of NCP remained as particulate organic carbon in the mixed layer of the iron-enriched patch, while 23% was lost by horizontal dispersion and 0–29% was exported. The closure of the carbon budget is hampered by the lack of measurements of dissolved organic carbon (DOC), by a major uncertainty in carbon export, and by use of empirical conversion factors in estimates of carbon biomass and metabolic rates. Lagrangian carbon-budget studies may be improved by direct measurement of all major carbon parameters and conversion factors. Carbon cycling in the SOIREE bloom resembled that in ‘natural’ algal blooms in the open Southern Ocean in some respects, but not in all. Daily NCP in the SOIREE bloom (70 mmol m⁻² d⁻¹) was higher than in natural blooms, partly because other studies did not account for horizontal dispersion, were for longer periods or included less productive areas. The build-up of POC stock and carbon export as a fraction of NCP in SOIREE were in the lower range of observations elsewhere.     

Coefficient of horizontal eddy diffusion of a tracer on the water surface as a function of the wave age

A relationship between the statistical parameters of horizontal diffusion and the parameters of the energy-containing part of the frequency spectrum of sea-surface elevations is found depending on the wave age Ω and the ratio between the wind speed at 10 m and the phase velocity of the peak of a wave. It has been observed in [1–7] that the diffusion coefficient K(r) of a patch of size r increases as r ^β, where 1.15 < β < 4/3, and the patch area S(t) increases with time as t ^γ, where 2 < γ < 3. As was calculated in [15], in the energy-containing part of the elevation frequency spectrum, S(ω) ～ ω^?n , where n = 13/3 for young waves with Ω > 2, n = 4 for waves with 1.2 < ψ < 2, and n = 11/3 for developed waves with 0.83 < Ω < 1.2. It is found that β = (n + 1)/4 and γ = 8/(7 ? n). These relations explain the entire set of observed exponents: β = 4/3 and γ = 3 for young waves and β = 1.15 and γ = 2.34 for large sizes (up to 1000 km) and times (up to a month) when it is found here that β = 7/6 and γ = 2.4.     

Horizontal refraction modal tomography of the ocean with modeinteractions

The acoustical tomography scheme for inferring a three-dimensional sound-speed field within some area based on the measurements of horizontal refraction angles is reviewed. The numerical simulations made so far were based on the assumption of the adiabaticity of low-frequency mode propagation. In this paper, an inversion scheme is presented that accounts for the acoustic mode interaction. We found that, generally, the interaction weakly affects the horizontal refraction angle, and it can be accounted for by iterations. Numerical simulations for the case of an Atlantic “meddy” corresponding to a strong double channel stratification are presented. Only three iterations were required to retrieve the exact strong sound-speed-field inhomogeneity within an area of 500 km×500 km     

Statistics from Lagrangian observations

We review statistical analyses of Lagrangian data from the ocean. These can be grouped into studies involving single particles and those with pairs or groups of particles. Single particle studies are the most common. The prevalent analysis involves binning velocities geographically to estimate the Eulerian means and lateral diffusivities. However single particle statistics have also been used to study Rossby wave propagation, the influence of bottom topography and eddy heat fluxes. Other studies have used stochastic models to simulate dispersion, calculated Lagrangian frequency spectra and examined the relation between Lagrangian and Eulerian integral scales. Studies involving pairs of particles are fewer, and the results are not well-established yet. There are indications that pair separations grow exponentially in time below the deformation radius, as is also the case in the stratosphere. The behavior at larger scales is less clear, indicating either a turbulent cascade or dispersion by the sheared large-scale circulation. In addition, three or more particles can be used to measure relative vorticity and divergence.     

Reducing horizontal diffusion errors in σ-coordinate coastal ocean models with a second-order Lagrangian-interpolation finite-difference scheme

When a steep bottom slope exists, it is well known that conventional methods for calculating horizontal diffusion in sigma-coordinate coastal ocean models causes spurious transport (e.g. salinity, temperature, and sediments) and currents. In this study, a second-order accurate finite-difference algorithm and program have been developed to reduce the spurious numerical diffusion errors. In the proposed algorithm, the finite differencing is performed in the x-z coordinate system to approximate the horizontal gradient. Each variable in the finite differential formation is calculated in the sigma-coordinate grid cells using a second-order Lagrangian interpolation polynomial. In conjunction with a stepwise bottom boundary condition, numerical experiments show that the proposed finite-difference scheme considerably reduces numerical errors compared to conventional approaches when dealing with horizontal diffusion over steep topography, which often occurs in coastal oceans and navigation channels.     

Verification of a parabolic equation for propagation of weakly-nonlinear waves

Verification tests of linear wave propagation models for cases of waves propagating through a refractive focus typically overpredict peak amplitudes in the vicinity of the focus. Using a parabolic equation model for the combined refraction-diffraction of weakly-nonlinear waves [valid for Ursell number U_r < O(1)], we show that, in a particular experiment, much of the discrepancy between laboratory measurements and linear wave model predictions can be accounted for by including the effect of nonlinearity, which is important in the region of the focus.     

Effect of divergence,stretching and nonnormal distribution on apparent diffusion of surface drifters

Kawai (1976, 1979) derived two simple relations among the apparent diffusivity (based on the geometric-average variance for principal axes of spreading) for many surface drifters in a patch and horizontal divergence and isotropic turbulent diffusivity for ambient water. By a new term medley diffusivity is meant an average of the products of the position coordinates relative to the centroid of drifters and the residual velocity, that is, the velocity left by subtracting the mean velocity and the velocity due to the linear velocity gradient from the velocity of a drifter. Regarding the medley diffusivity as the isotropic turbulent diffusivity that is weighted anisotropically according to drifters spreading along each of the principal axes, this paper derives a new relation with a form intermediate between the above two relations. Using the relation, this paper defines the new critical period within which the effect of horizontal divergence on the apparent diffusion surpasses that of isotropic turbulent diffusion, and discusses the time-scales on which the space-scale dependence of the squared divergence (Kawai, 1985b) is based. Stretching of a patch of drifters and deviation of drifters' position from the bivariate normal distribution enhance the effect of isotropic turbulent diffusion on the apparent diffusion. This stretching effect is equivalent to the shear effect on a horizontal plane. Remarks on reducing errors in the estimation of isotropic turbulent diffusivity are made.