Mellor, George L., March 2019: Comments on “A Wave-Resolving Simulation of Langmuir Circulations with a Nonhydrostatic Free-Surface Model: Comparison with Craik–Leibovich Theory and an Alternative Eulerian View of the Driving Mechanism”. Journal of Physical Oceanography, 49(3), DOI:10.1175/JPO-D-18-0222.1. Abstract
The results of the subject paper are reviewed wherein credible Langmuir cells are produced by a numerical solution of the primitive fluid dynamic equations with a free surface. Whereas it is a major achievement, the claim that the same results support the general application of the so-called vortex force equations is challenged.
Marsooli, R, P M Orton, and George L Mellor, July 2017: Modeling wave attenuation by salt marshes in Jamaica Bay, New York, using a new rapid wave model. Journal of Geophysical Research: Oceans, 122(7), DOI:10.1002/2016JC012546. Abstract
Using a new rapid-computation wave model, improved and validated in the present study, we quantify the value of salt marshes in Jamaica Bay – a highly urbanized estuary located in New York City – as natural buffers against storm waves. We improve the MDO phase-averaged wave model by incorporating a vegetation-drag-induced energy dissipation term into its wave energy balance equation. We adopt an empirical formula from literature to determine the vegetation drag coefficient as a function of environmental conditions. Model evaluation using data from laboratory-scale experiments show that the improved MDO model accurately captures wave height attenuation due to submerged and emergent vegetation. We apply the validated model to Jamaica Bay to quantify the influence of coastal-scale salt marshes on storm waves. It is found that the impact of marsh islands is largest for storms with lower flood levels, due to wave breaking on the marsh island substrate. However, the role of the actual marsh plants, Spartina alterniflora, grows larger for storms with higher flood levels, when wave breaking does not occur and the vegetative drag becomes the main source of energy dissipation. For the latter case, seasonality of marsh height is important; at its maximum height in early fall, S. alterniflora causes twice the reduction as when it is at a shorter height in early summer. The model results also indicate that the vegetation drag coefficient varies one order of magnitude in the study area, suggest exercising extra caution in using a constant drag coefficient in coastal wetlands.
Mellor, George L., September 2017: Reply to “Comments on ‘A Combined Derivation of the Integrated and Vertically Resolved, Coupled Wave–Current Equations'". Journal of Physical Oceanography, 47(9), DOI:10.1175/JPO-D-17-0096.1.
Mellor, George L., July 2016: On theories dealing with the interaction of surface waves and ocean circulation. Journal of Geophysical Research: Oceans, 121(7), DOI:10.1002/2016JC011768. Abstract
The classic theory for the interaction of surface gravity waves and the general ocean circulation entails the so-called wave radiation stress terms in the phase-averaged momentum equation. The equations of motion are for the combined Eulerian current and Stokes drift. On the other hand, a more recent approach includes the so-called vortex force term in the momentum equation wherein the only wave property is Stokes drift. The equations of motion are for the Eulerian current. The idea has gained traction in the ocean science community, a fact that motivates this paper. A question is: can both theories be correct? This paper answers the question in the negative and presents arguments in favor of the wave radiation theory. The vortex force approach stems from an interesting mathematical construct, but it does stand up to physical or mathematical scrutiny as described in this paper. Although not the primary focus of the paper, some discussion of Langmuir circulation is included since the vortex force was first introduced as the basis of this oceanic cellular phenomenon. Finaly the paper explains the difference in the derivation of the radiation stress theory and the vortex force theory: the later theory entails errors related to its use of curl and reverse-curl [or uncurl] processes.
Mellor, George L., June 2015: A Combined Derivation of the Integrated and Vertically Resolved, Coupled Wave-Current Equations. Journal of Physical Oceanography, 45(6), DOI:10.1175/JPO-D-14-0112.1. Abstract
There exist different theories representing the effects of surface gravity waves on oceanic flow fields. In the past, the author has conjectured that the vertically integrated, two-dimensional fluid equations of motion put forward by Longuet-Higgins and Stewart are correct and that theories that differ from their theory cannot be entirely correct; this paper explores these differences. Longuet-Higgins and Stewart deduced vertically integrated, two dimensional equations featuring a wave radiation stress term in the fluid dynamic, momentum equation. More recently, the author has proposed vertically dependent, three-dimensional equations which have required correction but which when vertically integrated, agreed with the earlier, two-dimensional equations. In this paper, we derive both vertically independent and vertically dependent equations from the same base and, importantly, using the same expression for pressure in the belief that the paper will contribute to the understanding and clarification of this seemingly difficult topic in ocean dynamics. An error in the classical papers by Longuet-Higgins and Stewart has been detected. Although the final phase-averaged result was correct, the error has had consequences in the development of vertically dependent equations. The prognostic equations in this paper are for the Eulerian current plus Stokes drift; towards the end of the paper these equations are contrasted with prognostic equations for the Eulerian current alone.
Longuet-Higgins and Stewart (J Fluid Mech 13:481–504, 1962; Deep-Sea Res 11:529–562, 1964) and later Phillips (1977) introduced the problem of waves incident on a beach, from deep to shallow water. From the wave energy equation and the vertically integrated continuity equation, they inferred velocities to be Stokes drift plus a return current so that the vertical integral of the combined velocities was nil. As a consequence, it can be shown that velocities of the order of Stokes drift rendered the advective term in the momentum equation negligible resulting in a simple balance between the horizontal gradients of the vertically integrated elevation and wave radiation stress terms; the latter was first derived by Longuet-Higgins and Stewart. Mellor (J Phys Oceanogr 33:1978–1989, 2003a), noting that vertically integrated continuity and momentum equations were not able to deal with three-dimensional numerical or analytical ocean models, derived a vertically dependent theory of wave–circulation interaction. It has since been partially revised and the revisions are reviewed here. The theory is comprised of the conventional, three-dimensional, continuity and momentum equations plus a vertically distributed, wave radiation stress term. When applied to the problem of waves incident on a beach with essentially zero turbulence momentum mixing, velocities are very large and the simple balance between elevation and radiation stress gradients no longer prevails. However, when turbulence mixing is reinstated, the vertically dependent radiation stresses produce vertical velocity gradients which then produce turbulent mixing; as a consequence, velocities are reduced, but are still larger by an order of magnitude compared to Stokes drift. Nevertheless, the velocity reduction is sufficient so that elevation set-down obtained from a balance between elevation gradient and radiation stress gradients is nearly coincident with that obtained by the aforementioned papers. This paper includes four appendices. The first appendix demonstrates the numerical process by which Stokes drift is excluded from the turbulence stress parameterization in the momentum equation. A second appendix determines a bottom slope criterion for the application of linear wave relations to the derivation of the wave radiation stress. The third appendix explores the possibility of generalizing results by non-dimensionalization. The final appendix applies the basic theory to a problem introduced by Bennis and Ardhuin (J Phys Oceanogr 41:2008–2012, 2011).
Mellor, George L., October 2013: Pressure-slope momentum transfer in ocean surface boundary layers coupled with gravity waves. Journal of Physical Oceanography, 43(10), DOI:10.1175/JPO-D-13-068.1. Abstract
The paper focuses on the consequences of including surface and subsurface, wind forced, pressure-slope momentum transfer into the oceanic water column, a transfer process which competes with now-conventional turbulence transfer based on mixing coefficients. Horizontal homogeneity is stipulated as is customary when introducing a new surface boundary layer model or significantly new vertical momentum transfer physics to an existing model. An introduction to pressure-slope momentum transfer is first provided by a phase-resolved, vertically dependent analytical model which excludes turbulence transfer. Then there follows a discussion of phase-averaging; an appendix is an important adjunct to the discussion. Finally, a coupled wave-circulation model which includes pressure-slope and turbulence momentum transfer is presented and numerically executed. The calculated temperatures compare well with measurements from ocean weather station papa.
There are differences in the literature concerning
the vertically dependent equations that couple currents and
waves. In this paper, currents are purposely omitted until
the end. Isolating waves from currents allows one to focus
on two main topics: an explanation of Stokes drift with
apparent mean vorticity obtained from an otherwise
irrotational flow and the determination of vertically
dependent wave radiation stress which, when vertically
integrated, conforms to that obtained by Longuet-Higgins
and Stewart (1964) and Phillips (1977) nearly 50 years ago
and, more recently, by Smith (2006). Discussion begins
with the simple case of nonlinear flow beneath a stationary
wavy wall.
In response to the comments of Ardhuin et al., the formulation of Mellor has been revised. Solutions of the model equations are now consistent with known deep-water behavior and agree with the shallow water, analytical–numerical experiment put forward by Ardhuin et al.
Mellor, George L., M A Donelan, and Leo Oey, October 2008: A surface wave model for coupling with numerical ocean circulation models. Journal of Atmospheric and Oceanic Technology, 25(10), DOI:10.1175/2008JTECHO573.1. Abstract
A surface wave model is developed with the intention of coupling it to three-dimensional ocean circulation models. The model is based on a paper by Mellor wherein depth-dependent coupling terms were derived. To be compatible with circulation models and to be numerically economical, this model is simplified compared to popular third-generation models. However, the model does support depth and current refraction, deep and shallow water, and proper coupling with depth-variable currents.
The model is demonstrated for several simple scenarios culminating in comparisons of model calculations with buoy data during Hurricane Katrina and with calculations from the model Simulating Waves Nearshore (SWAN); for these calculations, coupling with the ocean was not activated.
This is a revision of a previous paper dealing
with three-dimensional wave-current interactions. It is shown that the
continuity and momentum equations in the absence of surface waves can
include waves after the addition of three-dimensional radiation stress
terms, a fairly simple alteration for numerical ocean circulation models.
The velocity that varies on time and space scales, which are large compared
to inverse wave frequency and wavenumber, is denoted by ûα
and, by convention, is called the “current.” The Stokes drift is labeled
uSα and the mean velocity is Uαûα + uSα. When
vertically integrated, the results here are in agreement with past
literature.
Surface wind stress is empirical, but transfer of the
stress into the water column is a function derived in this paper. The wave
energy equation is derived, and terms such as the advective wave velocity
are weighted vertical integrals of the mean velocity. The wave action
equation is not an appropriate substitute for the wave energy equation when
the mean velocity is depth dependent.
Mellor, George L., 2005: Some consequences of the three-dimensional current and surface wave equations. Journal of Physical Oceanography, 35(11), DOI:10.1175/JPO2794.1. Abstract
Three-dimensional, interacting current and surface gravity wave equations have recently been derived and compared with their counterpart vertically integrated equations; they are in the form of sigma-coordinate equations. The purpose of this paper is to examine some of the consequences of these equations including energy transfer between mean energy, wave energy, and turbulence energy, to frame some outstanding research issues, to provide a Cartesian version of the sigma-coordinate equations, and to compare with other formulations of wave–current interaction. In general, the paper is intended to set the stage for the development of numerical coupled surface wave and three-dimensional general circulation models. These models often include a flow-dependent turbulence-based viscosity.
Ezer, Tal, and George L Mellor, 2004: A generalized coordinate ocean model and a comparison of the bottom boundary layer dynamics in terrain-following and in z-level grids. Ocean Modelling, 6(3-4), DOI:10.1016/S1463-5003(03)00026-X. Abstract
Sensitivity studies with a new generalized coordinate ocean model are performed in order to compare the behavior of bottom boundary layers (BBLs) when terrain-following (sigma or combined sigma and z-level) or z-level vertical grids are used, but most other numerical aspects remain unchanged. The model uses a second-order turbulence closure scheme that provides surface and BBL mixing and results in a quite realistic climatology and deep water masses after 100 year simulations with a coarse resolution (1° × 1°) basin-scale terrain-following grid. However, with the same turbulence scheme but using a z-level grid, the model was unable to produce dense water masses in the deep ocean. The latter is a known problem for coarse resolution z-level models, unless they include highly empirical BBL schemes.
A set of dense water overflow experiments with high-resolution grids (10 and 2.5 km) are used to investigate the influence of model parameters such as horizontal diffusivity, vertical mixing, horizontal resolution, and vertical resolution on the simulation of bottom layers for the different coordinate systems. Increasing horizontal diffusivity causes a thinner BBL and a bottom plume that extends further downslope in a sigma grid, but causes a thicker BBL and limited downslope plume extension in a z-level grid. A major difference in the behavior of the BBL in the two grids is due to the larger vertical mixing generated by the turbulence scheme over the step-like topography in the z-level grid, compared to a smaller vertical mixing and a more stably stratified BBL in the sigma grid. Therefore, the dense plume is able to maintain its water mass better and penetrates farther downslope in the sigma grid than in the z-level grid. Increasing horizontal and vertical resolution in the z-level grid converges the results toward those obtained by a much coarser resolution sigma coordinate grid, but some differences remain due to the basic differences in the mixing process in the BBL.
Mellor, George L., and A Blumberg, 2004: Wave breaking and ocean surface layer thermal response. Journal of Physical Oceanography, 34(3), 693-698. Abstract PDF
The effect of breaking waves on ocean surface temperatures and surface boundary layer deepening is investigated. The modification of the Mellor–Yamada turbulence closure model by Craig and Banner and others to include surface wave breaking energetics reduces summertime surface temperatures when the surface layer is relatively shallow. The effect of the Charnock constant in the relevant drag coefficient relation is also studied.
Burnett, W H., I V Kamenkovich, A I Gordon, and George L Mellor, 2003: The Pacific/Indian Ocean pressure difference and its influence on the Indonesian Seas circulation: Part I—The study with specified total transports. Journal of Marine Research, 61(5), 577-611. Abstract PDF
The main objective of this paper is to investigate the overall balance of momentum and energy within the Indonesian Seas to better understand the factors that control the total transport of the Indonesian Throughflow. Two models are used in the investigation: a "first-step" heuristic channel model and a more sophisticated "second-step," barotropic numerical model that incorporates high-resolution coastline and bottom topography. The experiments show that the barotropic model develops typical horizontal circulation patterns for the region. An analysis of the overall momentum and energy balances suggests that the total transport of the Indonesian Throughflow does not depend exclusively on the inter-ocean pressure difference but on other factors, including local winds, bottom form stresses, and the resultant of pressure forces acting on the internal sides.
Kamenkovich, I V., W H Burnett, A I Gordon, and George L Mellor, 2003: The Pacific/Indian Ocean pressure difference and its influence on the Indonesian Seas circulation: Part II—The study with specified sea-surface heights. Journal of Marine Research, 61(5), 613-634. Abstract PDF
In Part II we construct new numerical solutions to further analyze our results in Part I (Burnett et al., 2003), that indicate the lack of a unique relationship between the Pacific/Indian Ocean pressure difference and the total transport of the Indonesian Throughflow (ITF). These new solutions involve perturbations of the sea level relative to the original solutions. We present detailed analyses of the overall momentum and energy balances for these new solutions to stay consistent with the procedures developed in Part I. The results validate our conclusions regarding the lack of a unique relationship between the pressure head and the value of the total transport of the ITF. However, based on results from all the experiments, we have found that the seasonal variations of the total transport of the ITF are in phase with the pressure-head variations. Thus the hypothesis by Wyrtki (1987) that the pressure head, measured by the sea-surface-height difference between Davao (Philippines) and Darwin (Australia), is well correlated with the total transport is qualitatively supported.
Lee, Hyun-Chul, and George L Mellor, June 2003: Numerical simulation of the Gulf Stream and the deep circulation. Journal of Oceanography, 59(3), DOI:10.1023/A:1025520027948343-357. Abstract
The Gulf Stream system has been numerically simulated with relatively high resolution and realistic forcing. The surface fluxes of the simulation were obtained from archives of calculations from the Eta-29 km model which is an National Center for Environment Prediction (NCEP) operational atmospheric prediction model; synoptic fields are available every 3 hour. A comparison between experiments with and without surface fluxes shows that the effect of the surface wind stress and heat fluxes on the Gulf Stream path and separation is closely related to the intensification of deep circulations in the northern region. Additionally, the separation of the Gulf Stream and the downslope movement of the Deep Western Boundary Current (DWBC) are reproduced in the model results. The model DWBC crosses under the Gulf Stream southeast of Cape Hatteras and then feeds the deep cyclonic recirculation east of the Bahamas. The model successfully reproduces the cross-sectional vertical structures of the Gulf Stream, such as the asymmetry of the velocity profile, and this structure is sustained along the downstream axis. The distribution of Root Mean Square (RMS) elevation anomaly of the model shows that the eddy activity of the Gulf Stream is realistically reproduced by the model physics. The entrainment of the upper layer slope current into the Gulf Stream occurs near cross-over; the converging cross-stream flow is nearly barotropic.
Lee, Hyun-Chul, and George L Mellor, February 2003: Numerical simulation of the Gulf Stream System: The Loop Current and the deep circulation. Journal of Geophysical Research, 108(C2), 3043, DOI:10.1029/2001JC001074. Abstract
The Loop Current and the deep circulation in the Gulf of Mexico are numerically investigated by a primitive equation, sigma coordinate ocean model with realistic surface fluxes obtained from an atmospheric forecast model. A deep cyclonic circulation, bounded by the deep basin in the eastern Gulf of Mexico, is spun up by the Loop Current; the deep cyclonic circulation is coincident with a southward current of the Loop Current eastern limb and weakens after Loop Current ring separation and cessation of the southward current. The anticyclonic, semienclosed Loop Current also induces anticyclonic lower layer columnar eddies in the eastern gulf. These lower layer eddies decouple from the upper layer Loop Current. The westward translation speed of a Loop Current ring is about 2.16-5.18 km d-1; the lower layer eddies have a higher speed and lead the rings into the central gulf. The time-averaged surface circulation of the Gulf of Mexico basin is anticyclonic, mainly because of the transport of anticyclonic vorticity by Loop Current rings in the surface layer an average lower layer cyclonic circulation occurs along the continental slope of the basin.
Mellor, George L., 2003: Comments on "Stability of algebraic non-equilibrium second-order closure models" by H. Burchard and E. Deleersnjder [Ocean Modelling 3 (2001) 33–50]. Ocean Modelling, 5(2), 193-194. PDF
Surface wave equations appropriate to three-dimensional ocean models apparently have not been presented in the literature. It is the intent of this paper to correct that deficiency. Thus, expressions for vertically dependent radiation stresses and a definition of the Doppler velocity for a vertically dependent current field are obtained. Other quantities such as vertically dependent surface pressure forcing are derived for inclusion in the momentum and wave energy equations. The equations include terms that represent the production of turbulence energy by currents and waves. These results are a necessary precursor for three-dimensional ocean models that handle surface waves together with wind- and buoyancy-driven currents. Although the third dimension has been added here, the analysis is based on the assumption that the depth dependence of wave motions is provided by linear theory, an assumption that is the basis of much of the wave literature.
Mellor, George L., 2002: Oscillatory bottom boundary layers. Journal of Physical Oceanography, 32(11), 3075-3088. Abstract PDF
A turbulence closure model is applied to the case of an oscillating boundary layer; model calculations compare favorably with data. Wave-induced oscillations can be temporally resolved in a one-dimensional model but not in three-dimensional ocean models, and, indeed, statistical wave models, working in consort with ocean models, can only provide information on expected wave periods and amplitudes. Therefore, in this paper, a way has been found to parameterize the effects of bottom flow oscillations; it entails augmenting the turbulence shear production as a function of amplitude and period of the oscillation, the bottom shear stress of the mean current flow, and the angle between the directions of the oscillations and the mean flow. The more conventional method of solving for an apparent wall roughness is also investigated in an appendix.
Mellor, George L., S Häkkinen, Tal Ezer, and R Patchen, 2002: A generalization of a sigma coordinate ocean model and an intercomparison of model vertical grids In Ocean Forecasting: Conceptual Basis and Applications, Pinardi, N., and J. D. Woods, Eds., Springer-Verlag, 55-72.
Zedler, S E., T D Dickey, Scott C Doney, J F Price, X Yu, and George L Mellor, 2002: Analyses and simulations of the upper ocean's response to Hurricane Felix at the Bermuda testbed mooring site: 13-23 August 1995. Journal of Geophysical Research, 107(C12), DOI:10.1029/2001JC000969. Abstract PDF
The center of Hurricane Felix passed 85 km to the southwest of the Bermuda Testbed Mooring (BTM; 31°44´N, 64°10´W) site on 15 August 1995. Data collected in the upper ocean from the BTM during this encounter provide a rare opportunity to investigate the physical processes that occur in a hurricane's wake. Data analyses indicate that the storm caused a large increase in kinetic energy at near-inertial frequencies, internal gravity waves in the thermocline, and inertial pumping, mixed layer deepening, and significant vertical redistribution of heat, with cooling of the upper 30 m and warming at depths of 30–70 m. The temperature evolution was simulated using four one-dimensional mixed layer models: Price-Weller-Pinkel (PWP), K Profile Parameterization (KPP), Mellor-Yamada 2.5 (MY), and a modified version of MY2.5 (MY2). The primary differences in the model results were in their simulations of temperature evolution. In particular, when forced using a drag coefficient that had a linear dependence on wind speed, the KPP model predicted sea surface cooling, mixed layer currents, and the maximum depth of cooling closer to the observations than any of the other models. This was shown to be partly because of a special parameterization for gradient Richardson number (Rg5PP) shear instability mixing in response to resolved shear in the interior. The MY2 model predicted more sea surface cooling and greater depth penetration of kinetic energy than the MY model. In the MY2 model the dissipation rate of turbulent kinetic energy is parameterized as a function of a locally defined Richardson number (RgMY2) allowing for a reduction in dissipation rate for stable Richardson numbers (RgMY2) when internal gravity waves are likely to be present. Sensitivity simulations with the PWP model, which has specifically defined mixing procedures, show that most of the heat lost from the upper layer was due to entrainment (parameterized as a function of bulk Richardson number RbPWP), with the remainder due to local Richardson number (RgPWP) instabilities. With the exception of the MY model the models predicted reasonable estimates of the north and east current components during and after the hurricane passage at 25 and 45 m. Although the results emphasize differences between the modeled responses to a given wind stress, current controversy over the formulation of wind stress from wind speed measurements (including possible sea state and wave age and sheltering effects) cautions against using our results for assessing model skill. In particular, sensitivity studies show that MY2 simulations of the temperature evolution are excellent when the wind stress is increased, albeit with currents that are larger than observed. Sensitivity experiments also indicate that preexisting inertial motion modulated the amplitude of poststorm currents, but that there was probably not a significant resonant response because of clockwise wind rotation for our study site.
Mellor, George L., 2001: One-dimensional, ocean surface layer modeling: A problem and a solution. Journal of Physical Oceanography, 31(3), 790-809. Abstract PDF
The first part of this paper is generic: it demonstrates a problem associated with one-dimensional, ocean surface layer model comparisons with ocean observations. Unlike three-dimensional simulations or the real ocean, kinetic energy can inexorably build up in one-dimensional simulations, which artificially enhances mixing. Adding a sink term to the momentum equations counteracts this behavior. The sink term is a surrogate for energy divergence available to three-dimensional models but not to one-dimensional models.
The remainder of the paper deals with the Mellor-Yamada boundary layer model. There exists prior evidence that the model's summertime surface temperatures are too warm due to overly shallow mixed layer depths. If one adds a sink term to approximate three-dimensional model behavior, the warming problem is exacerbated, creating added incentive to seek an appropriate model change. Guided by laboratory data, a Richardson-number-dependent dissipation is introduced and this simple modification yields a favorable improvement in the comparison of model calculations with data even with the momentum sink term in place.
Blaha, J P., G H Born, N L Guinasso, Jr, H J Herring, G A Jacobs, F J Kelly, R R Leben, R D Martin, Jr, and George L Mellor, et al., 2000: Gulf of Mexico ocean monitoring system. Oceanography, 13(2), 10-17.
Burnett, W H., V M Kamenkovich, D A Jaffe, A L Gordon, and George L Mellor, 2000: Dynamical balance in the Indonesian Seas circulation. Geophysical Research Letters, 27(17), 2705-2708. Abstract PDF
A high resolution, four-open port, non-linear, barotropic ocean model (2D POM) is used to analyze the Indonesian Seas circulation. Both local and overall momentum balances are studied. It is shown that geostrophy holds over most of the area and that the Pacific-Indian Ocean pressure difference is essentially balanced by the resultant of pressure forces acting on the bottom.
Burnett, W H., V M Kamenkovich, George L Mellor, and A L Gordon, 2000: The influence of the pressure head on the Indonesian Seas circulation. Geophysical Research Letters, 27(15), 2273-2276. Abstract PDF
A high resolution, regional, non-linear, barotropic ocean model (2D POM) was used to show that a pressure difference between the Pacific and Indian Ocean does not significantly influence the total transport of the Indonesian throughflow.
Ezer, Tal, and George L Mellor, 2000: Sensitivity studies with the North Atlantic sigma coordinate Princeton Ocean Model. Dynamics of Atmospheres and Oceans, 32(3-4), 185-208. Abstract PDF
The sigma coordinate, Princeton Ocean Model (POM) has been configured for the North Atlantic Ocean between 5°N and 50°N as part of data assimilation, model predictability and intercomparison studies. The model uses a curvilinear orthogonal grid with higher resolution in the western North Atlantic and lower resolution in the eastern North Atlantic. A series of experiments, each one of a 10-year duration, are performed to evaluate the sensitivity of the ocean mean state and variability to model parameters and model configuration; these experiments include open vs. closed boundary conditions, low vs. high resolution grids, and different choices of diffusion and viscosity. The results show that the use of closed boundaries together with near-boundary buffer zones where temperature and salinity are relaxed towards the observed values give less realistic flows, weaker recirculation gyres and less realistic Gulf Stream separation than do open boundary conditions. The experiments show that the sensitivity of the ocean variability in the model to the choice of the Smagorinsky diffusion and viscosity coefficients significantly differs from one region to another and largely depends on other attributes such as the mean position of the Gulf Stream in each simulation. A 50% change in model resolution in the Gulf Stream region has a larger effect on ocean variability than a change of diffusivity by a factor of 10. In areas where either the high or the low resolution models have sufficient resolution, as in the Gulf of Mexico, they are able to produce variability comparable to that observed from altimeter data; elsewhere, model variability is underestimated.
Mellor, George L., 1999: Comments on "On the Utility and Disutility of JEBAR". Journal of Physical Oceanography, 29(8), 2117-2118. PDF
Mellor, George L., Leo Oey, and Tal Ezer, 1998: Sigma coordinate pressure gradient errors and the seamount problem. Journal of Atmospheric and Oceanic Technology, 15(5), 1122-1131. Abstract
In a recent paper by Mellor, et al., it was found that, in two-dimensional (x, z) applications with finite horizontal viscosity and zero diffusivity, the velocity error, associated with the evaluation of horizontal density or pressure gradients on a sigma coordinate grid, prognostically disappeared, leaving behind a small and physically insignificant distortion in the density field. The initial error is numerically consistent in that it decreases as the square of the grid increment size. In this paper, we label this error as a sigma error of the first kind.
In three-dimensional applications, the authors have encountered an error that did not disappear and that has not been understood by us or, apparently, others. This is a vorticity error that is labeled a sigma error of the second kind and is a subject of this paper. Althogh it does not prognostically disappear, it seems to be tolerably small. To evaluate these numerical errors, the authors have adopted the seamount problem initiated by Beckman and Haidvogel. It represents a stringent test case, as evidenced by their paper, wherein the model is initialized with horizontal isopycnals, zero velocity, and no forcing; then, any velocities that develop must be considered errors.
Two appendices are important adjuncts to the paper, the first providing theoretical confirmation and understanding of the numerical results, and the second delving into additional errors related to horizontal or isosigma diffusion. It is, however, shown that satisfactory numerical solutions are obtained with zero diffusivity.
Ezer, Tal, and George L Mellor, 1997: Data assimilation experiments in the Gulf Stream region: How useful are satellite-derived surface data for nowcasting the subsurface fields?Journal of Atmospheric and Oceanic Technology, 14(6), 1379-1391. Abstract PDF
Satellite-derived surface data have become an important source of information for studies of the Gulf Stream system. The question of just how useful these datasets are for nowcasting the subsurface thermal fields, however, remains to be fully explored. Three types of surface data--sea surface temperature (SST), sea surface height (SSH), and Gulf Stream position (GSP)--are used here in a series of data assimilation experiments to test their usefulness when assimilated into a realistic primitive equation model. The U.S. Navy's analysis fields from the Optimal Thermal Interpolation System are used to simulate the surface data and to evaluate nowcast errors. Correlation factors between variations of the surface data and variations of the subsurface temperature are used to project the surface information into the deep ocean, using data and model error estimates and an optimal interpolation approach to blend model and observed fields.
While assimilation of each surface data source shows some skill in nowcasting the subsurface fields (i.e., reducing errors compared to a control case without assimilation), SSH data reduce errors more effectively in mid-depths (around 500 m), and SST data reduce errors more effectively in the upper layers (above 100 m). Assimilation of GSP is effective in nowcasting the deep Gulf Stream, while the model dynamics produce eddies that are not included in the GSP analysis. An attempt to optimally combine SST and SSH data in the asimilation shows an improved skill at all depths compared to assimilation of each set of data separately.
Ezer, Tal, and George L Mellor, 1997: Simulations of the Atlantic Ocean with a free surface sigma coordinate ocean model. Journal of Geophysical Research, 102(C7), 15,647-15,657. Abstract PDF
A sigma coordinate, free surface numerical model with turbulence dynamics has been implemented for the Atlantic Ocean and the Greenland Sea, from 80°S to 80°N. It is driven at the surface by monthly mean sea surface temperature and wind stress climatologies and is executed for 30 years. This is the first time that a model of this type, previously used mostly for coastal and regional simulations, has been implemented for the entire Atlantic Ocean and run for a long period of time. The model horizontal circulation, thermohaline overturning circulation, and meridional heat fluxes are described; the results are compared with observations and the results of other models. The model produces intense deep western boundary currents and complicated gyre structures associated with small-scale topographic variations. The meridional overturning circulation consists of about 14 Sv (1 Sv = 106 m3 s-1) of southward flowing deep water mass crossing the equator and a northward heat flux with a maximum value of more than 1 PW (1015 W). Although the maximum meridional heat flux is comparable to estimates obtained from observations, the amplitude of the seasonal variations of northward heat flux across 26°N is underestimated in comparison with observations; it is similar to that obtained by other models.
Aikman III, F, George L Mellor, and Tal Ezer, et al., 1996: Towards an operational nowcast/forecast system for the U.S. East Coast In Modern Approaches to Data Assimilation in Ocean Modeling, The Netherlands, Elsevier Science Publishers, 347-376. Abstract
A model system consisting of the Princeton ocean model forced by forecast surface fluxes of momentum and heat from the regional atmospheric Eta model is at the heart of the East Coast Ocean Forecast System. Existing near-real-time data sets, including coastal water level gauge data and satellite-derived sea surface temperature and altimetry data, are being used operationally for model evaluation purposes and ultimately for assimilation into the ocean model. The first twelve months of comparisons between 24-hour forecasted and observed subtidal coastal water levels indicate a meridional average correlation coefficient of 0.65, an rms difference of 10 cm, and shows that the forecasts represent over 60% of the observed subtidal variability. A number of sensitivity experiments are underway and a series of enhancements are soon to be implemented, including modification of the surface heat and momentum fluxes; the inclusion of atmospheric pressure loading, riverine fresh water and surface fresh water (evaporation and precipitation) fluxes, and tidal forcing; and accounting for the effects of thermal expansion and contraction. In order to evaluate and improve the basic ocean model and system, the implementation of data assimilation is currently being withheld, however, data assimilation methodologies have been developed and the sea surface temperature and altimeter data currently available in near-real-time will be used for these purposes.
Mellor, George L., 1996: Introduction to Physical Oceanography, Woodbury, NY: American Institute of Physics, 260 pp.
Mellor, George L., and X-H Wang, 1996: Pressure compensation and the bottom boundary layer. Journal of Physical Oceanography, 26(10), 2214-2222. Abstract PDF
It is an observed characteristic of oceans that velocities and horizontal pressure gradients are larger near the ocean surface than they are in deeper water. This is conventionally labeled "pressure compensation" whereby baroclinic structure, comprising sloping isopycnal surfaces, is adjusted so that surface pressure gradients are reduced in deeper water. In this paper, a two-dimensional flow in a channel is numerically modeled to demonstrate the baroclinic adjustment process and its relationship to the bottom boundary layer. A simple analytical model is also developed and defines the timescale of the adjustment process.
Ezer, Tal, George L Mellor, and R J Greatbatch, 1995: On the interpentadal variability of the North Atlantic Ocean: Model simulated changes in transport, meridional heat flux and coastal sea level between 1955-1959 and 1970-1974. Journal of Geophysical Research, 100(C6), 10,559-10,566. Abstract PDF
Previous studies by Greatbatch, et al. (1991) indicate significant changes in the North Atlantic thermohaline structure and circulation between the pentads 1955-1959 and 1970-1974, using data analyzed by Levitus (1989a,b,c) and a simple diagnostic model by Mellor, et al. (1982). In this paper, these changes are modeled using a three-dimensional, free surface, coastal ocean model. Diagnostic and short-term prognostic calculations are used to infer the dynamically adjusted fields corresponding to the observed hydrographic and wind stress climatology of each pentad. While the results agree with earlier studies indicating that the Gulf Stream was considerably weaker (by about 30 Sv) during the 1970s compared to the 1950s, they also indicate some changes in the poleward heat transport, although the statistical significance of these changes relative to sampling errors in the data is not clear. The change of wind pattern between the two pentads, associated with changes in sea surface temperature, resulted in changes in the Ekman contribution to the poleward heat flux transport. The modeled sea level along the North American coast shows a sea level rise of about 5-10 cm between 1955-1959 and 1970-1974; a comparison with observed sea level at 15 tide gage stations shows good agreement. Most of the coastal sea level change is attributed to changes in thermohaline ocean circulation and wind stress; thermal expansion seems to play a lesser role. The methodology tested here demonstrates an effective way to estimate climate changes in ocean circulation and sea level from observed hydrographic data and winds using ocean models to enhance and analyze the data.
Mellor, George L., and Tal Ezer, 1995: Sea level variations induced by heating and cooling: An evaluation of the Boussinesq approximation in ocean models. Journal of Geophysical Research, 100(C10), 20,565-20,577. Abstract PDF
In this paper, a sigma coordinate ocean model is modified to remove the commonly used Boussinesq approximation so that the effect of thermal expansion is exactly included in the basic equations in order to cope with the seasonal heating cycle and the detection of climate change through variation in sea level height. Tests are performed to evaluate the differences between Boussinesq and non-Boussinesq calculations under different heating and cooling conditions and different model domains. For an idealized case of a flat bottom, shallow ocean basin without wind forcing, simulations of a warm eddy show that the non-Boussinesq dynamics have only a minor effect on the baroclinic current field. However, vertically averaged velocities, though small compared with the baroclinic velocities, are cyclonic for the Boussinesq calculation and anticyclonic for the non-Boussinesq calculation. The results indicate that global or closed basin Boussinesq models should be able to simulate most of the observed steric sea level changes on seasonal or climate timescales, when corrected by a spatially uniform, time-dependent factor calculated from the volume-averaged density change. The seasonal variation of the globally averaged sea level calculated from climatological data is small, about 1 cm. Variations in steric sea level in regional models, both Boussinesq and non-Boussinesq, may differ from those of global models owing to the unknown transport across their boundaries associated with the local heating and cooling. A spatially uniform, time-dependent correction, similar to that associated with thermal expansion, is proposed to account for transport across open boundaries of regional models. Variations of sea level obtained from a Boussinesq model of the Atlantic Ocean approximate the seasonal signal due to the heating/cooling cycle of each hemisphere as observed by satellite altimeter data.
Schwab, D J., W P O'Connor, and George L Mellor, 1995: On the net cyclonic circulation in large stratified lakes. Journal of Physical Oceanography, 25(6), 1516-1567. Abstract PDF
This paper proposes a possible explanation for the mean cyclonic circulation in large stratified lakes. The condition of no heat flux through the bottom boundary causes the isotherms to dip near the shores to intersect the sloping bottom orthogonally. This "doming" of the thermocline causes an internal pressure gradient in the surface layer with higher pressure nearshore and results in a geostrophic cyclonic circulation.
Zavaterelli, M, and George L Mellor, 1995: A numerical study of the Mediterranean Sea circulation. Journal of Physical Oceanography, 25(6), 1384-1414. Abstract PDF
A primitive equation ocean model that makes use of a curvilinear orthogonal grid and a sigma-coordinate system was used to simulate the Mediterranean Sea. The model was forced with monthly climatological values of wind stress, heat, and salinity flux. With the help of the curvilinear horizontal grid, the larger scales of the entire Mediterranean Sea are modeled, and the topography around the narrow and shallow Straits Gibraltar is also reasonably well represented. The resulting model inflow and outflow seems to mimic the real Mediterranean, often in considerable detail. Levantine Intermediate Water is formed in the Levantine Basin and exits through the Strait of Sicily and the Strait of Gibraltar. Deep-water formation processes are clearly represented by the model.
The model results indicate that in the western Mediterranean the wind stress is very important in establishing the summer northward shift of the Atlantic inflow. Lateral boundary runoff, surface salinity, and heat fluxes are necessary for the maintenance of the cyclonic circulation in the northern Balearic Basin and enhance the seasonal reversal of the circulation in the Tyrrhenian Sea. An interesting result is the existence of a seasonal variation in the path of the Levantine Intermediate Water.
Ezer, Tal, and George L Mellor, 1994: Continuous assimilation of GEOSAT altimeter data into a three-dimensional primitive equation Gulf Stream model. Journal of Physical Oceanography, 24(4), 832-847. Abstract PDF
A three-dimensional data assimilation scheme is described and tested, using the GEOSAT altimeter data and a high-resolution, primitive equation, numerical ocean model of the Gulf Stream region. The assimilation scheme is based on an optimal interpolation approach in which data along satellite tracks are continuously interpolated horizontally and vertically into the model grid and assimilated with the model prognostic fields. Preprocessed correlations between surface elevation anomalies and subsurface temperature and salinity anomalies are used to project surface information into the deep ocean; model and data error estimates are used to optimize the assimilation. Analysis fields derived from the Navy's Optimum Thermal Interpolation System are used to initialize the model and to provide some estimate of errors. To evaluate the effectiveness of the assimilation scheme, the errors of model oceanic fields (surface elevation, Gulf Stream axis, temperature) with data assimilation are compared with errors without data assimilation (i.e., a pure forecast). Although some mesoscale meanders and rings are not well produced by the assimilation model, consistent reduction of errors by the assimilation is demonstrated. The vertical distribution of errors reveals that the scheme is most effective in nowcasting temperatures at mid-depth (around 500 m) and less effective near the surface and in the deep ocean. The scheme is also more effective in nowcasting the Gulf Stream axis location than in nowcasting temperature variations. A comparison of the assimilation scheme during two periods shows that the nowcast skill of the assimilated model is reduced in May-September 1988, compared to May-July 1987, due to poor coverage of the altimeter data during 1988. This paper is one step toward a dynamic model and data assimilation system, which when fully developed, should provide useful nowcast and forecast information.
Ezer, Tal, and George L Mellor, 1994: Diagnostic and prognostic calculations of the North Atlantic circulation and sea level using a sigma coordinate ocean model. Journal of Geophysical Research, 99(C7), 14,159-14,171. Abstract PDF
The North Atlantic circulation and sea surface height, determined from hydrographic and wind stress data are calculated with a free surface, primitive equation ocean model. The model grid, a vertical sigma - coordinate and horizontal curvilinear orthogonal system, makes it possible to resolve coastal regions with complicated topography that were unresolved in previous calculations. When running in a diagnostic mode, in which the temperature and the salinity fields are fixed and equal to the Levitus annual mean fields, the vertically integrated North Atlantic circulation is very similar to though more detailed than that obtained from previous calculations using simpler diagnostic models. On the other hand, the meridional, zonally averaged flows and the poleward heat transport from the purely diagnostic calculations are noisy and unrealistic. However, short prognostic calculations of only 30 days following the diagnostic run allow intensification of the western boundary current and removal of noise due to inconsistencies between the hydrographic data and bottom topography and produce a more realistic meridional circulation and poleward heat transport, with a maximum value of 1.2 x 1015 W which is comparable to estimates based on observations. The calculated sea level along the North American continent agrees with previous estimates but provides more spatial detail. Analysis of the dynamic adjustment process shows that this process is dominated by the effect of bottom topography through the action of the bottom pressure torque. This study is a first step in applying a realistic numerical model, previously used primarily for estuaries and coastal regions, to basin-scale-ocean problems.
Ezer, Tal, George L Mellor, S Häkkinen, and C Mauritzen, 1994: Simulations of the Arctic and North Atlantic Oceans In The Atlantic Climate Change Program, Proceedings from the principal investigators meeting, NOAA, University Corporation for Atmospheric Research, 115-119. Abstract
Numerical simulations of the North Atlantic and the Arctic Oceans were and are being carried out using the Princeton Ocean Model (POM; Blumberg and Mellor, 1987). POM uses a bottom-following sigma-coordinate system and a coastal-following curvilinear orthogonal grid; it has been developed primarily to study estuaries and coastal regions, but is increasingly being applied to basin-scale problems and climate studies.
To assist this brief discussion we refer to model grids illustrated in Figures 1a,b,c,d. Grid (a) refers mainly to accomplished work as described in the "Background" section. Grids (b), (c) and (d) represent "Current Progress"; one or more of these grids will factor into future "Plans."
Mellor, George L., Tal Ezer, and Leo Oey, 1994: The pressure gradient conundrum of sigma coordinate ocean models. Journal of Atmospheric and Oceanic Technology, 11(4), 1126-1134. Abstract
Much has been written of the error in computing the horizontal pressure gradient associated with sigma coordinates in ocean or atmospheric numerical models. These also exists the concept of "hydrostatic inconsistency" whereby, for a given horizontal resolution, increasing the vertical resolution may not be numerically convergent.
In this paper, it is shown that the differencing scheme cited here, though conventional, is not hydrostatically inconsistent; the sigma coordinate, pressure gradient error decreases with the square of the vertical and horizontal grid size. Furthermore, it is shown that the pressure gradient error is advectively eliminated after a long time integration. At the other extreme, it is shown that diagnostic calculations of the North Atlantic Ocean using rather coarse resolution, and where the temperature and salinity and the pressure gradient error are held constant, do not exhibit significant differences when compared to a calculation where horizontal pressure gradients are cmputed on z-level coordinates. Finally, a way of canceling the error ab initio is suggested.
Willems, R C., Tal Ezer, and George L Mellor, et al., 1994: Experiment evaluates ocean models and data assimilation in the Gulf Stream. EOS, 75(34), 385, 391, 394.
Ezer, Tal, George L Mellor, D-S Ko, and Z Sirkes, 1993: A comparison of Gulf Stream sea surface height fields derived from GEOSAT altimeter data and those derived from sea surface temperature data. Journal of Atmospheric and Oceanic Technology, 10(1), 76-87. Abstract PDF
Two types of satellite data, GEOSAT altimeter data and sea surface temperature data (SST), are compared and evaluated for their usefulness in assimilation into a numerical model of the Gulf Stream region. Synoptic sea surface height (SSH) fields are derived from the SST data in the following way: first, three-dimensional temperature and salinity analysis fields are obtained through the Optimum Thermal Interpolation System (OTIS), and then SSH fields are calculated using a primitive equation, free-surface, numerical model running in a diagnostic mode. The aforementioned SSH fields are compared with SSH fields obtained from the GEOSAT altimeter data. Use of GEOSAT data requires an estimate of the mean SSH field relative to the Earth geoid. Three different methods to obtain the mean SSH field are demonstrated. The first method uses altimetry and SST data; the second uses a diagnostic calculation with climatological data; and the third uses prognostic numerical calculations. The three estimates compared favorably with each other and with estimates obtained elsewhere. The comparison of the synoptic SSH fields derived from both data types reveals similarity in the Gulf Stream meanders and some mesoscale features, but shows differences in strength of eddies and in variability far from the Gulf Stream. Due to the smoothed nature of the OTIS analysis fields, the SSH derived from altimetry data has larger variability amplitudes compared to that derived from SST data. The statistical interpolation method, which is used to interpolate altimetry data from satellite tracks onto the model grid is also evaluated for its filtering effect and its sensitivity to different parameters. The SSH variability of the Gulf Stream was calculated from two years of the exact repeat mission of the GEOSAT satellite, where altimeter data were interpolated daily onto the model grid. It is suggested here that some of the underestimation of mesoscale variations by statistical interpolation methods, as indicated by previous studies, may be explained by the filtering effect of the scheme.
Jewell, P W., R F Stallard, and George L Mellor, 1993: Numerical studies of bottom shear stress and sediment distribution on the Amazon continental shelf. Journal of Sedimentary Petrology, 63(4), 734-745. Abstract
The relation between bottom shear stress and the distribution of bottom sediments on the Amazon continental shelf has been studied using a three-dimensional, primitive-equation computer model that incorporates the turbulence-closure scheme of Mellor and Yamada (1982) for calculating eddy diffusivity and a simple algorithm for computing nonlinear wave-current influences on bottom shear stress. Model results compare reasonably well with salinity data sets for the Amazon plume. Model results on distribution of bottom currents and bottom shear stresses help explain some of the observed sedimentological features of the Amazon continental shelf. High concentrations of suspended sediment in the Amazon River are transported outward over the continental shelf and northward by the North Brazil Coastal Current. As this sediment settles out of the water column, it forms the prograding, subaqueous delta described by Nittrouer, et al. (1986). Accumulation rates are greatest shoreward of the 40-m isobath due to a zone of convergent, cross-shelf residual tidal velocities. Little sediment is deposited in the shallow parts of the shelf, where bottom shear stress exceeds 10 dynes/cm2 over a diurnal tidal cycle. Zones of laminated sand and mud on the Amazon continental shelf coincide with areas of high interseasonal differences in bottom shear stress. Our results suggest that our model may be useful in interpreting sedimentation in ancient sedimentary basins as well.
Oey, Leo, and George L Mellor, 1993: Subtidal variability of estuarine outflow, plume, and coastal current: A model study. Journal of Physical Oceanography, 23(1), 164-171. Abstract
The time evolution of an estuary plume and its coastal front over a continental shelf is numerically calculated here using a three-dimensional model with eddy mixing based on the turbulence kinetic energy closure. The plume and front system is found to be unsteady with a natural period of about 5-10 days, during which the plume pulsates and intermittent coastal currents propagate down the coast.
Xue, H, and George L Mellor, 1993: Instability of the Gulf Stream front in the South Atlantic Bight. Journal of Physical Oceanography, 23(11), 2326-2350. Abstract PDF
To understand Gulf Stream meanders in the South Atlantic Bight, the growth of three-dimensional perturbations along two-dimensional frontal zones is examined by using linearized primitive equations. The Fourier-Galerkin method and the orthogonal collocation method are combined to formulate the spectral model. Emphasis is placed on the effects of cross-frontal topographic slope on the stability of the front, and on the characteristics of the most unstable modes. Attention is directed to the cross sections upstream and downstream of the Charleston Bump, which is a topographic feature near 31°N. The major results obtained from this linear study are that 1) the growth rate of the most unstable mode decreases and the associated phase speed increases after incorporating cross-front topographic gradients; 2) the most unstable solution found in the region downstream of the Charleston Bump has a slightly longer wavelength and slower phase speed than those found in the region upstream of the Bump.
Ezer, Tal, D-S Ko, and George L Mellor, 1992: Modeling and forecasting the Gulf Stream. Marine Technology Society Journal, 26(2), 5-14. Abstract
Numerical simulations are performed to evaluate the forecast skill of a model of the Gulf Stream system. The model is a high resolution (eddy resolving) coastal ocean model, which includes thermohaline dynamics and a turbulence scheme to provide vertical mixing coefficients. In a series of forecast experiments, the model is initialized with synoptic temperature and salinity fields obtained from satellite observations and the U.S. Navy's Optimum Thermal Interpolation System (OTIS). It then calculates the forecast fields (e.g., temperature, salinity, and sea surface height) for the next two weeks. In the three cases presented here, the model forecst gave a better estimate of the ocean than did persistence (i.e., the assumption of no change), showing a forecast skill for at least two months.
Sensitivity studies demonstrate the effects of vertical grid resolution, horizontal diffusion, and smoothing on the forecast skill of the model when compared to OTIS fields. The forecast skill is improved when the vertical grid is refined and when smoothing or horizontal diffusion is large enough to remove small-scale spatial variations from the forecast fields; such variations are missing from the smoothed OTIS fields but may exist in the real ocean.
The study shows that numerical models can be used to aid commercial and navy operations in forecasting oceanic fields; nevertheless, there are still deficiencies in numerical models and difficulties in quantitative evaluation of the forecast skill of ocean models due to the sparse coverage of oceanic measurements.
Ezer, Tal, and George L Mellor, 1992: A numerical study of the variability and the separation of the Gulf Stream, induced by surface atmospheric forcing and lateral boundary flows. Journal of Physical Oceanography, 22(6), 660-682. Abstract PDF
A primitive equation regional model is used to study the effects of surface and lateral forcing on the variability and the climatology of the Gulf Stream system. The model is an eddy-resolving, coastal ocean model that includes thermohaline dynamics and a second-order turbulence closure scheme to provide vertical mixing. The surface forcing consists of wind stress and heat fluxes obtained from the Comprehensive Ocean-Atmosphere Data Set (COADS). Sensitivity studies are performed by driving the model with different forcing (e.g., annual versus zero surface forcing or monthly versus annual forcing). The model climatology, obtained from a five-year simulation of each case, is then compared to observed climatologies obtained from satellite-derived SST and hydrocast data.
The experiments in which surface heat flux and wind stress were neglected show less realistic Gulf Stream separation and variability, compared with experiments in which annual or seasonal forcing are used. A similar unrealistic Gulf Stream separation is also obtained when the slope-water inflow at the northeast boundary is neglected. The experiments suggest that maintaining the density structure and the concommitant geostrophic flow in the northern recirculation gyre plays an important role in the separation of the Gulf Stream. The maintenance of the recirculation gyre is affected by heat transfer, wind stress, and slope-water inflow. The heat transfer involves several processes: lateral eddy transfer, surface heat flux, and vertical mixing. Further improvement of the Gulf Stream separation and climatology are obtained when seasonal changes in the lateral temperature and salinity boundary conditions are included.
The seasonal climatology of the model calculations compare reasonably well with the observed climatology. Although total transports on open boundaries are maintained at climatological values, there are, nevertheless, large seasonal and spatial variations of Gulf Stream transport between Cape Hatteras and 62°W. These changes are accompanied by transport changes in the northern recirculation gyre.
Häkkinen, S, and George L Mellor, 1992: Modeling the seasonal variability of a coupled arctic ice-ocean system. Journal of Geophysical Research, 97(C12), 20,285-20,304. Abstract
Results from modeling studies of the ice-ocean system in the Arctic Basin and in the Norwegian-Greenland-Barents seas are presented. We used a three-dimensional coupled ice-ocean model developed at Princeton University. The ocean model applies the primitive equations and a second moment turbulence closure for turbulent mixing. The snow-ice model uses a three-level thermodynamic scheme which resembles Semtner's (1976a) model. Our conclusions based on the seasonal simulations are as follows. 1) Using monthly climatological surface heat flux and wind stress, the seasonal variability of the ice cover is quite realistic in that the thickest ice is located north of Greenland and the average ice thickness is about 3 m. The largest deviation between the simulated and observed ice cover is in the Greenland Sea where oceanic conditions determine the ice edge. Basically, the monthly climatological forcing does not result in strong enough mixing to bring sufficient heat from the deep ocean to keep the central Greenland Sea gyre ice free. The results improve for both the ice cover and ocean by invoking daily wind forcing for which we first chose year 1987. In the ocean model, the large mixing events associated with storm passages are resolved, and as a result, the overall oceanic structure in the Greenland Sea appears to be more realistic. However, no deep convection takes place in the model during 1987 which is likely the result of diminished storm activity in the northern part of the Greenland Sea. The ice thickness field appears to be very anomalous 1987, so an experiment with 1986 daily wind forcing was also done, which resulted in an ice thickness field similar to some reported from other ice models. 2) Both monthly and daily surface forcing result in a similar behavior of the Atlantic waters in the Arctic Basin. The Atlantic waters circulate at about the observed level, between 400 and 600 m. The survival of the Atlantic waters in the basin depends strongly on the heat loss through the ice cover, and it appears that too much heat is lost on the Eurasian side through the ice because the simulated Atlantic waters are too cool by about 0.2-0.5°C. 3) For the monthly climatology case, a large amount of cold and salty water enters the Eurasia Basin from the Kara and Laptev seas area and finds its way toward the Canada Basin. This water mass appears to result from ice formation in the Kara and Laptev seas. When applying the daily forcing, this deep salinity maximum disappears due to increased mixing on the shelves. Nevertheless, this suggests a mechanism within the Arctic Ocean as to why the deep Canada Basin is much saltier than the Eurasia Basin.
Häkkinen, S, George L Mellor, and L H Kantha, 1992: Modeling deep convection in the Greenland Sea. Journal of Geophysical Research, 97(C4), 5389-5408. Abstract
The development of deep convective events in the high-latitude ocean is studied using a three-dimensional, coupled ice-ocean model. Oceanic mixing is described according to the level 2.5 turbulence closure scheme in which convection occurs in a continuous way, i.e., convective adjustment is not invoked. The model is forced by strong winds and surface cooling. Strong upwelling at the multiyear ice edge and consequent entrainment of warm Atlantic waters into the mixed layer is produced by winds parallel to the ice edge. Concomitant cooling drives deep convection and produces chimney-like structures. Inclusion of a barotropic mean flow over topography to the model provides important preconditioning and selects the location of deep convection. The most efficient preconditioning occurs at locations where the flow ascends a slope. In a stratified environment similar to the Greenland Sea with a 12 m s-1 wind the model simulations show that localized deep convection takes place after about 10 days to depths of 1000 m.
Oey, Leo, Tal Ezer, George L Mellor, and P Chen, 1992: A model study of "bump" induced western boundary current variabilities. Journal of Marine Systems, 3, 321-342. Abstract
A time-dependent, three-dimensional numerical model is used to study the effects of a bottom irregularity or "bump" on western boundary current (WBC) variabilities along a simplified shelf and slope. Numerical experiments with (i) no bottom bump, (ii) a small bump and (iii) a large bump have been conducted. Case (i) produces low variabilities and cases (ii) and (iii) show significant increase in slope and shelf energetics both downstream and upstream of the bump. Disturbances generated at the bump are well correlated with flow variabilities upstream. Downstream variabilities are caused by meander development following the WBC deflection by the bump, while topographic waves excite upstream variabilities. The model also indicates two modes of deflection paths, small- and large-amplitude paths, downstream of the bump. These findings are further supported by results obtained from a Gulf Stream simulation which incorporates the bathymetry of the U.S. South Atlantic Bight, and which has a more realistic boundary forcing. The simulated eddy kinetic energy distribution shows three regions of variability which are of interest: one inshore (and slightly downstream) and one offshore of the Charleston Bump, and a third region over the shelfbreak some 150-200 km upstream of the Bump. The inshore and offshore maxima are due to the small and large amplitude deflection paths of the model Gulf Stream, respectively, while the upstream maximum is presumably due to topographic wave activity.
Mellor, George L., 1991: An equation of state for numerical models of oceans and estuaries. Journal of Atmospheric and Oceanic Technology, 8, 609-612. PDF
Mellor, George L., and Tal Ezer, 1991: A Gulf Stream model and an altimetry assimilation scheme. Journal of Geophysical Research, 96(C5), 8779-8795. Abstract PDF
A continuous data assimilation scheme and a multilayer, primitive equation, numerical model are described. The model is an eddy-resolving, coastal ocean model that has been extended to include the Gulf Stream region. It has complete thermohaline dynamics, a bottom-following, sigma, vertical coordinate system, and a coastal-following, curvilinear orthogonal, horizontal coordinate system. Calculated model fields are used to provide a model climatology and correlations between subsurface temperature and salinity anomalies and surface elevation anomalies. An optimal interpolation method, the surface to subsurface correlations, and estimated model and data errors are the basis of the assimilation technique. Altimetry anomaly data extracted from the model calculations according to the GEOSAT orbital schedule are used to test the assimilation scheme and to provide nowcasts and forecasts. Sensitivity studies are performed to test the effects of various parameters of the scheme. It is found that the scheme is less efficient in the shallow continental shelf area than in the deeper regions of the model. The results show significant nowcast skill, with area-averaged rms error for surface elevation and subsurface properties of about 40-50% of the corresponding error of the unassimilated case. Good forecast skill, better than persistence, is demonstrated for 10-20 days; there is little skill after 30-40 days. Increasing the density of the satellite altimetry data (especially by decreasing the separation distance between tracks) should decrease the nowcast rms error to about 15% and improve the forecast.
Galperin, B, and George L Mellor, 1990: A time-dependent, three-dimensional model of the Delaware Bay and river system. Part 1: Description of the model and tidal analysis. Estuarine, Coastal and Shelf Science, 31, 231-253. Abstract
A three-dimensional, time-dependent numerical model is used to simulate the dynamics and thermodynamics of Delaware Bay, River and adjacent continental shelf. This study describes the first attempt to model an estuary and the contiguous shelf as a coupled hydrodynamic and thermodynamic system. Here, in Part 1, a description of the model, boundary conditions and forcing information is provided. Numerical results are compared with surface elevation data at several locations throughout the Bay and the River, as well as with the observations collected by the National Ocean Service during their 1984-85 circulatory study. It is shown that a vertically-integrated, two-dimensional version of the model predicts realistic amplitudes but with some phase error. The full three-dimensional model reduces the phase error but underpredicts the tidal range; this is due to the higher values of horizontal viscosity required by the three-dimensional model. The model accounts for non-linear, shallow-water effects and reproduces the observed amplification of the high-frequency tidal components from the mouth of the Bay to the head of the River at Trenton.
Galperin, B, and George L Mellor, 1990: A time-dependent, three-dimensional model of the Delaware Bay and river system. Part 2: Three-dimensional flow fields and residual circulation. Estuarine, Coastal and Shelf Science, 31, 255-281. Abstract
The three-dimensional model of Delaware Bay, River and adjacent continental shelf was described in Part 1. Here, Part 2 of this two-part paper demonstrates that the model is capable of realistic simulation of current and salinity distributions, tidal cycle variability, events of strong mixing caused by high winds and rapid salinity changes due to high river runoff. The 25-h average subtidal circulation strongly depends on the wind forcing. Monthly residual currents and salinity distributions demonstrate a classical two-layer estuarine circulation wherein relatively low salinity water flows out at the surface and compensating high salinity water from the shelf flows at the bottom. The salinity intrusion is most vigorous along deep channels in the Bay. Winds can generate salinity fronts inside and outside the Bay and enhance or weaken the two-layer circulation pattern.
Since the portion of the continental shelf included in the model is limited, the model shelf circulation is locally wind-driven and excludes such effects as coastally trapped waves and interaction with Gulf Stream rings; nevertheless, a significant portion of the coastal elevation variaibility is hindcast by the model. Also, inclusion of the shelf improves simulation of salinity inside the Bay compared with simulations where the salinity boundary condition is specified at the mouth of the Bay.
Häkkinen, S, and George L Mellor, 1990: One hundred years of Arctic ice cover variations as simulated by a one-dimensional, ice-ocean model. Journal of Geophysical Research, 95(C9), 15,959-15,969. Abstract PDF
A one-dimensional ice-ocean model consisting of a second moment, turbulent closure, mixed layer model and a three-layer snow-ice model has been applied to the simulation of Arctic ice mass and mixed layer properties. The results for the climatological seasonal cycle are discussed first and include the salt and heat balance in the upper ocean. The coupled model is then applied to the period 1880-1985, using the surface air temperature fluctuations from Hansen et al. (1983) and from Wigley at al. (1981). The analysis of the simulated large variations of the Arctic ice mass during this period (with similar changes in the mixed layer salinity) shows that the variability in the summer melt determines to a high degree the variability in the average ice thickness. The annual oceanic heat flux from the deep ocean and the maximum freezing rate and associated nearly constant minimum surface salinity flux did not vary significantly interannually. This also implies that the oceanic influence on the Arctic ice mass is minimal for the range of atmospheric variability tested.
Mellor, George L., B J Herring, and R Patchen, 1990: The Princeton/Dynalysis ocean model oceanic circulation models In Coastal Ocean Prediction Systems Program: Understanding and Managing Our Coastal Ocean, Vol. 2. New Hampshire, Institute for the Study of Earth, Oceans, and Space, 77-113. Abstract
The history of the Princeton/Dynalysis General Circulation model begins with the development of analytical turbulence closure models of small-scale turbulence at Princeton University (1973, 1974, 1977) such that vertical mixing or the inhibition of vertical mixing of momentum, temperature and salinity (or any other ocean property) can be predicted with considerable confidence. The history of the model is sketched and its properties are summarized. Presented are applications of the prognostic three-dimensional circulation model and its various submodels for various investigative purposes: New York Harbor, Delaware Bay, Middle Atlantic Bight, the entire Atlantic coast Continental Shelf, the Santa Barbara Channel, and the California coastal shelf.
Galperin, B, Anthony Rosati, L H Kantha, and George L Mellor, 1989: Modeling rotating stratified turbulent flows with application to oceanic mixed layers. Journal of Physical Oceanography, 19(7), 901-916. Abstract PDF
Rotational effects on turbulence structure and mixing are investigated using a second-moment closure model. Both explicit and implicit Coriolis terms are considered. A general Criterion for rotational effects to be small is established in terms of local turbulent Rossby numbers. Characteristic length scales are determined for rotational effects and Monin-Obukhov type similarity theory is developed for rotating stratified flows. A one-dimensional version of the closure model is then applied to simulate oceanic mixed layer evolution. It is shown that the effects of rotation onmixed layer depth tend to be small because of the influence of stable stratification. These findings contradict a hypothesis of Garwood et al. that rotational effects on turbulence are responsible for the disparity in the mixed-layer depths between the eastern and western regions of the equatorial Pacific Ocean. The model is also applied to neutrally stratified flows to demonstrate that rotation can either stabilize or destabilize the flow.
Kantha, L H., and George L Mellor, 1989: A numerical model of the atmospheric boundary layer over a marginal ice zone. Journal of Geophysical Research, 94(C4), 4959-4970. Abstract PDF
A two-dimensional, multilevel model for simulating changes in the atmospheric boundary layer across a marginal ice zone is described and applied to off-ice, on-ice, and along-ice edge wind conditions. The model incorporates a second-moment closure for parameterizing the intensification and suppression of turbulent mixing in the boundary layer due to stratification effects. For off-ice winds, as the atmospheric boundary layer passes from cold smooth ice onto warm open water, the onset of intense convection raises the inversion. Over the transition zone of rough rafted ice with open leads, the shear stress on the ice cover increases significantly before dropping down to the downstream values over water. Such nonmonotonic surface stress could be the cause of divergence of sea ice near the ice edge in a marginal ice zone. These results are in agreement with the one-layer model simulations of off-ice winds by Overland et al. (1983). For on-ice wind conditions, as the warm flow in the boundary layer encounters the cold ice conditions, the resulting stable stratification could rapidly suppress the turbulence in the boundary layer, leading to the development of a shallow inversion and an associated jet. When the wind is predominantly along the ice edge, the temperature contrast between the open water and the ice could produce a thermal front at the ice edge in the boundary layer with strong associated turbulence. More observations are needed to verify these model predictions. Nevertheless, these model results suggest that it is important to account for the changes in the characteristics of the atmospheric boundary layer across the marginal ice zone in our attempts to understand the behavior of the ice cover in these regions.
Kantha, L H., and George L Mellor, 1989: Two-dimensional coupled ice-ocean model of the Bering Sea marginal ice zone. Journal of Geophysical Research, 94(C8), 10,921-10,935. Abstract PDF
A two-dimensional coupled ice-ocean model has been formulated and applied to the wintertime Bering Sea marginal ice zone. The oceanic component is a multi- level model that incorporates second-moment closure for turbulent mixing in the water column. The ice cover is modelled as a viscous-plastic continuum. Melting at the ice-ocean interface is computed by using well-known law-of-the- wall concepts in a turbulent boundary layer, with particular attention to the disparate momentum and scalar transfer resistance coefficients over rough walls. The thermodynamic and dynamical interactions between the ocean and the ice cover and the energy balances at the air-ice and air-sea interfaces are modelled according to the companion paper (Mellor and Kantha, this issue). The model incorporates barotropic tides, both diurnal and semidiurnal, for application to the Bering Shelf. Double-diffusive fluxes across the interface between the colder, fresher layer beneath the melting ice and the warmer, more saline water underneath are prescribed from laboratory data on double-diffusive convection. During winter, sea ice in the central Bering Sea is transported toward the shelf break by off-ice winds, where it encounters northward flowing warmer north Pacific waters and melts. It is this situation to which the two-dimensional model has been applied by neglecting all variations in the along-ice-edge direction. The water conditions downstream of the ice edge, the ice conditions upstream, and the wind stress are the primary inputs to the model. The model simulates transition from ice-covered to open ocean conditions and the associated ice edge front and the two-layer circulation underneath the ice cover. Sensitivity studies indicate that the density structure and the circulation beneath the ice and the position of the ice edge are rather sensitive to the parameters affecting the dynamics and the thermodynamics of the coupled ice-ocean system. Even small changes in the relevant parameters can cause a substantial retreat or advance of the ice edge, which may help explain why marginal ice zones are such dynamically active regions.
Mellor, George L., 1989: Retrospect on oceanic boundary layer modeling and second moment closure In Parameterization of Small-Scale Processes, 5th 'Aha Huliko'a Hawaiian Workshop, Hawaii, Hawaii Institute of Geophysics, 251-272. Abstract
Models that describe ocean surface and bottom layers are discussed in the context of numerical ocean modeling; second moment turbulence models are emphasized. Here, we deal with the methodology and with errors that have been encountered. Errors relate to: (i) inadequate model resolution and boundary conditions which are, oftentimes, severely filtered, (ii) errors in the surface forcing variables, and (iii) inherent errors in model physics. It is difficult to quantify and even identify the errors in each category. We begin here to evaluate (i). One finding is that error is reduced if observed, large horizontal scale, internal wave energy is included in the turbulence energy, closure equations.
Mellor, George L., and L H Kantha, 1989: An ice-ocean coupled model. Journal of Geophysical Research, 94(C8), 10,937-10,954. Abstract PDF
An ice model, an ocean model, and a method of coupling the models are described. The ice model is a synthesis, with variations and extensions, of previous modeling ideas. Ice thickness, concentration, velocity, and internal energy are prognostic variables. The ice thermodynamics are represented by temperatures at the snow surface, ice surface, the interior, and the bottom surface. Melting and freezing rates are calculated at the ice-atmosphere, ice-ocean, and atmosphere-ocean interfaces. A prescribed portion of summer meltwater can be stored on the surface and refrozen in the fall. The ocean model includes a second moment, turbulence closure submodel and enables one to solve for oceanic heat flux, the interfacial stress, and subsurface properties. In this paper the model is applied to one-dimensional simulations, but the equations are cited in a form for implementation by two- and three-dimensional models. In a companion paper (Kantha and Mellor, this issue) the model is used for two-dimensional (vertical plane) simulations in the Bering Sea. Several one-dimensional sensitivity studies are performed in the case where the ice model is decoupled from the ocean; here the oceanic heat flux and sea surface temperature are prescribed constants. The studies reveal the role and sensitivity of surface trapped meltwater, ice concentration, and ice divergence. With the coupled ice-ocean model, the seasonally varying oceanic heat flux and mixed layer properties are determined by the model. Some comparisons with observations in the central Arctic Ocean are possible. The role of the molecular sublayer immediately adjacent to the ice is examined; frazil ice production is related to the large disparity in the molecular diffusivities for temperature and salinity. The mixed layer model contains empirical constants which are known from turbulence data. The molecular sublayer parameterization requires one empirical parameter b, which is uncertain but, from this study, is assuredly greater than zero, the value implicit in previous models. The ice model requires the empirical parameters phiF and phiM to quantitatively account for freezing or melting processes in open leads; their values are also uncertain, but we present reasoning and sensitivity studies to suggest specific values. Finally, an empirical parameter G is introduced; it is the ratio of the value of the ice thickness used to represent average ice volume in the dynamic and thermodynamic equations to the value of the thickness needed in the heat conduction equation. Estimates of G are made from observed thickness distribution functions; sensitivity studies show it to be an important parameter.
Steele, Michael, George L Mellor, and M G McPhee, 1989: Role of the molecular sublayer in the melting or freezing of sea ice. Journal of Physical Oceanography, 19(1), 139-147. Abstract PDF
In an earlier paper, a second-moment turbulence closure model was applied to the problem of the dynamic and thermodynamic interaction of sea ice and the ocean surface mixed layer. An overly simplistic parameterization of the molecular sublayers of temperature and salinity within the mixed layer was used. This paper investigates the use of a more recent parameterization by Yaglom and Kader, which is supported by laboratory data. A relatively low melt rate results in the case where ice overlays warm water. This agrees with some recent observations in the interior of the marginal ice zone. A surface heat sink drives the freezing case which, because of the large differences in heat and salt molecular diffusivities, produces a strong supercooling effect. This is converted into an estimate of frazil ice production through a simple scheme. The model results provide an explanation for high frazil ice concentrations observed in the Arctic and Antarctic.
Rajkovic, B, and George L Mellor, 1988: Coastal ocean response to atmospheric forcing In International Colloquium on Ocean Hydrodynamics, 19th - Small-Scale Turbulence and Mixing on the Ocean, The Netherlands, Elsevier Science Publishers, 141-149. Abstract
In order to examine the response of a coastal ocean to atmospheric forcing, successive integrations of 2-D atmospheric and oceanic models are performed. The atmospheric model has a prescribed sea surface temperature that is independent of time and a prescribed, time-dependent land surface temperature. The oceanic model is forced by the wind obtained from the atmospheric model. In the case of constant sea surface temperature, the wind stress distribution is fairly constant in the cross-shore direction except in the vicinity of the coastline. With a sea surface temperature distribution corresponding to a well-developed upwelling situation, the atmosphere model develops a wind stress distribution with a pronounced decrease in a 40 km band next to the coast. The model ocean, forced with the wind stress obtained from the atmospheric run with homogeneous sea surface temperature, develops a strong upwelling zone and a strong equatorward current with an embedded jet near the coast. Forced with the wind stress from the atmospheric run with inhomogeneous sea surface temperature, the ocean run has a much weaker upwelling and a double structured alongshore current with poleward flow in the vicinity of the coastline and equatorward flow in the region away from the coast.
Mellor, George L., 1986: Numerical simulation and analysis of the mean coastal circulation off California. Continental Shelf Research, 6(6), 689-713. Abstract
A two-dimensional numerical model is applied to a coastal ocean wherein alongshore elevation and density gradients, normally calculated by a three-dimensional model, are instead supplied by climatologically averaged data for the California Current System between 25 and 40 degrees N. Surface wind stress is also obtained from climatological data. Both surface and bottom boundary layers are resolved in the model calculations; a second moment turbulence closure submodel supplies vertical diffusivities. Near steady state solutions are possible when surface buoyancy flux is imposed at the surface.
Model results are as follows: Southward wind stress produces a broad equatorward current with an embedded coastal jet in accordance with previous studies. Positive wind stress curl reduces the jet current and produces a poleward undercurrent which then surfaces as the curl is increased. The jet currents are reduced and poleward flow increases as bottom steepness increases; to a lesser extent, inclusion of the beta effect has a similar effect. The existence of near bottom, poleward or equatorward flow is explained rather simply in terms of the bottom stress resulting from the alongshore balance of surface wind stress and vertically integrated pressure gradient, the latter involving the alongshore surface elevation and density gradient. A further finding is that the upwelling circulation associated with wind stress is confined to the top 200 to 300 m of the ocean along the California coast.
Mellor, George L., M G McPhee, and Michael Steele, 1986: Ice-seawater turbulent boundary layer interaction with melting or freezing. Journal of Physical Oceanography, 16(11), 1829-1846. Abstract PDF
A second-moment, turbulence closure model is applied to the problem of the dynamic and thermodynamic interaction of sea ice and the ocean surface mixed layer. In the case of ice moving over a warm, ocean surface layer, melting is intrinsically a transient process; that is, melting is rapid when warm surface water initially contacts the ice. Then the process slows when surface water is insulated from deeper water due to the stabilizing effect of the melt water, and the thermal energy stored in the surface layer is depleted. Effectively, the same process prevails when ocean surface water flows under stationary ice in which case, after an initial rapid increase, the melting process decreases with downstream distance. Accompanying the stabilizing effect of the melt water is a reduction in the ice-seawater interfacial shear stress. This process and model simulations are used to explain field obervations wherein ice near the marginal ice zone diverges from the main pack.
When the surface ice layer is made to grow by imposing heat conduction through the ice, the surface ocean layer is destabilized by brine rejection and mixing in the water column is enhanced. The heat flux into the water column is a small percentage of the heat conduction through the ice.
Mellor, George L., 1985: Ensemble average, turbulence closure. Advances in Geophysics, 28B, 345-358.
Mellor, George L., and A Blumberg, 1985: Modeling vertical and horizontal diffusivities with the Sigma Coordinate System. Monthly Weather Review, 113(8), 1379-1383. Abstract PDF
The use of diffusive terms in numerical ocean models is examined relative to different coordinate systems. The conventional model for horizontal diffusion is found to be incorrect when bottom topographical slopes are large. A new formulation is suggested which is simpler than the conventional formulation when transformed to a sigma coordinate system and makes it possible to model realistically both surface Ekman and bottom boundary layers.
Mellor, George L., Leo Oey, R I Hires, and B Galperin, 1985: Three-dimensional numerical models for hindcasting or forecasting estuarine tides, currents and salinities In Applications of Real-time Oceanographic Circulation Modeling: Symposium Proceedings, Washington, DC, Marine Technology Society, 211-225.
Oey, Leo, George L Mellor, and R I Hires, 1985: A three-dimensional simulation of the Hudson-Raritan estuary. Part I: Description of the model and model simulations. Journal of Physical Oceanography, 15(12), 1676-1692. Abstract
A time-dependent, three-dimensional, finite difference simulation of the Hudson-Raritan estuary is presented. The calculation covers July-September 1980. The model estuary is forced by time-dependent observed winds, tidal elevation at open boundaries, and river and sewage discharges. Turbulence mixing coefficients in the estuary are calculated according to a second-moment, turbulence-closure submodel. Horizontal diffusivities are zero in the simulation and small-scale eddies produced by the interaction of unsteady, three-dimensional velocity and salinity fields with coastline and bottom bathymetry were resolved by the model. These eddies are important physical elements in shear dispersion processes in an estuary.
Model results show unstably stratified water columns produced by advection of waters of different densities. These instabilities produce intense mixing with verical eddy diffusivities reaching 2-3 times their neutral values. They occur most frequently at slack currents, during initial stages of flooding currents and also during up-estuary wind events. These three-dimensional, time-dependent solutions extend previous analytical model results and are consistent with observations in partially mixed and well mixed estuaries.
Model results show large subtidal response of velocity and salinity fields to wind forcing. Wind forcing modifies the density-induced flows in deep channels in the estuary and also the horizontal circulation in Raritan Bay where the average water depth is less than 5 m and tidal currents are weak.
Oey, Leo, George L Mellor, and R I Hires, 1985: A three-dimensional simulation of the Hudson-Raritan estuary. Part II: Comparison with observation. Journal of Physical Oceanography, 15(12), 1693-1709. Abstract
Results from time-dependent, three-dimensional numerical simulation of the Hudson-Raritan estuary are compared with observations. The comparison includes: 1) instantaneous salinity contours across a transect in the estuary; 2) amplitudes and phases of tidal constituents at four tide gauge and five current meter stations; 3) mean currents at nine meter locations, and mean salinity in the Hudson River; 4) kinetic energy spectra; and 5) response to wind forcing of subtidal current at an observational station near the mouth of the estuary.
Observations confirm the model's prediction of existence of density advection instabilities induced by differential advection of the three-dimensional density field. These instabilities produce intense vertical mixing and should significantly modify dispersion processes in the estuary. Effects of neap-spring tides on vertical stratifications are also simulated by the model. Simulated M2 phases at three tide gauge stations show improvement over the M2 phases obtained from a two-dimensional, vertically integrated tidal model. The improvement is presumably due to bottom boundary layer resolution and, therefore, improved representation of bottom friction in the three-dimensional model. Simulated (instantaneous and mean) currents compare reasonably well with observations, except at narrow channel regions where the model's resolution is inadequate. Simulated "density-induced" mean currents are weaker than those observed, a discrepancy attributed to neglect of temperature variations in the model. Horizontal diffusion coefficients are null in this model. The burden of horizontal dispersion is generally handled well by the model's adequate resolution of small-scale advective processes, as suggested by the model's correct simulation of the k-3 transfer spectrum law at high wavenumber k. In narrow rivers that are modeled two-dimensionally (x, z), the estimate of the horizontal dispersion due to vertical variabilities in velocity and salinity appears to be correct; however, mixing by lateral variability is absent so that the saline intrusion is somewhat underpredicted. At the mouth of the estuary, simulated subtidal current responses to wind forcing generally agree with observed responses. The response is partly barotropic, which is a result of balance between bottom friction, sea level setup from the adjacent continental shelf and wind stress, modified by local vertical velocity shears and baroclinic responses.
Oey, Leo, George L Mellor, and R I Hires, 1985: A three-dimensional simulation of the Hudson-Raritan estuary. Part III: Salt flux analyses. Journal of Physical Oceanography, 15(12), 1711-1720. Abstract
Salt fluxes and volume transports in an estuary vary considerably over subtidal time scales of a few days to weeks in response to wind and neap-spring tidal forcings. Results from a numerical simulation of the Hudson-Raritan estuary are used to study subtidal variations of salt fluxes and the physical mechanisms for salt balance in the estuary. Simulated salt fluxes are compared with available observations. Observations support the model's finding that analysis of volume and salt fluxes based on short-length data records (<30 days) can lead to misleading conclusions.
"Tidal trapping" effects due to coastline irregularities contribute most to the salt balance at the Sandy Hook-Rockaway Point transect and at the Narrows. A two-week observational record is analyzed to support this finding. Simulated subtidal variation of the tidal trapping term at the Sandy Hook-Rockaway Point transect compares well with that observed. In the Raritan Bay, where tidal currents are weak and effects of winds are significant, contributions to salt balance from vertical velocity and salinity gradients are comparable to transverse contributions. This occurs despite the fact that surface-to-bottom salinity differences during the simulation period-a period of low freshwater flow-never exceed 0.5% throughout most regions of the bay. A two-dimensional, depth-integrated xy-t model, in which the horizontal dispersion coefficients are modeled empirically, may not perform well in this case.
Oey, Leo, George L Mellor, and R I Hires, 1985: Tidal modeling of the Hudson-Raritan Estuary. Estuarine, Coastal and Shelf Science, 20, 511-527. Abstract
Tidal flow characteristics in the Hudson-Raritan Estuary are studied with a two-dimensional, depth-averaged finite difference model. Rivers are modeled as one-dimensional channels with variable width and depth and are calculated as part of the two-dimensional calculations at no extra computational cost. An extensive comparison of numerical, tidal calculations with observational data than has previously appeared in the literature is presented. Computed velocity and tidal elevation fields compare well with observations. Comparison with observations at the Sandy Hook-Rockaway Point transect indicates that the barotropic tidal residual current contributes significantly to the overall steady circulation in the harbor. The residual current is mainly induced by the coastal geometry and bottom topography through the nonlinear inertia effects.
Domaradzki, J A., and George L Mellor, 1984: A simple turbulence closure hypothesis for the triple-velocity correlation functions in homogeneous isotropic turbulence. Journal of Fluid Mechanics, 140, 45-61. Abstract
A simple two-point closure scheme for homogeneous axisymmetric turbulence is developed. For the isotropic case it is essentially an eddy-viscosity assumption in real space for the Karman-Howarth equation. The eddy-viscosity function for large internal Reynolds numbers is derived from Kolmogoroff's 1941 theory. For moderate Reynolds numbers of order 102, approximately the same expression for the eddy-viscosity function is determined from experimental data. The resulting closed equation for the double-correlation function is solved numerically for both large and moderate Reynolds numbers, and the results are compared with experimental data. Self-similar solutions of the basic equation predict turbulent energy decay inversely proportional to time. It is shown that the departure from this 'initial-period decay law' observed in laboratory data is due to the behavior of grid-produced correlation functions for large separation distances.
Blumberg, A, and George L Mellor, 1983: Diagnostic & Prognostic numerical circulation studies of the South Atlantic Bight. Journal of Geophysical Research, 88(C8), 4579-4592. Abstract
Some of the results from a series of diagnostic and prognostic numerical simulations of the circulation in the South Atlantic Bight (SAB) are described. The numerical model developed for the study is a three-dimensional, primitive equation, time dependent, delta coordinate model with an imbedded, turbulent closure submodel which should yield realistic Ekman surface and bottom layers. An implicit numerical scheme in the vertical direction and a mode-splitting technique in time are adopted for computational efficiency. A significant portion of the paper is concerned with realistic specification of initial conditions for temperature and salinity, surface forcing, and lateral open boundary conditions. The latter are determined by a simple diagnostic (geostrophic and Ekman dynamics) model which provides dynamically consistent temperature, salinity, and velocity boundary conditions. It appears from an examination of the numerical simulations that the full model yields results that share many features in common with our general understanding of the circulation of the South Atlantic Bight; the region includes shallow shelf waters as well as deeper water dominated by the Gulf Stream. Data for synoptic skill assessment, however, are not available.
Blumberg, A, B J Herring, L H Kantha, and George L Mellor, 1983: A prognostic California Shelf circulation model. EOS, 64(45), 727.
Mellor, George L., 1983: The coastal ocean, upwelling boundary layer. EOS, 64(45), 726.
Oey, Leo, and George L Mellor, 1983: Real time, 3-D simulation of the New York Harbor. EOS, 64(45), 744.
Mellor, George L., C R Mechoso, and E Keto, 1982: A diagnostic calculation of the general circulation of the Atlantic Ocean. Deep-Sea Research, Part I, 29(10A), 1171-1192. Abstract
The general circulation of the Atlantic Ocean is calculated from the low Rossby number equations of motions where all frictional terms except surface wind stress are neglected. Climatological sea surface wind stress, temperature, and salinity, and bottom topography are inputs to the calculation. Total transport is calculated by integrating the vertically integrated equations of motion along contours of constant planetary potential vorticity, f/H, where f is the Coriolis parameter and H is the local depth. The integration begins on the eastern boundary of the ocean where the total transport is assumed to be zero. Results are also obtained for three transport components; Ekman transport, thermohaline transport, and bottom velocity transport.
Despite the rather crude 1° x 1° calculation grid, the results show considerable detail, particularly in the higher latitude Atlantic Ocean. The calculations yield about a 25 x 106 m3 s-1 transport through the Straits of Florida increasing to a maximum Gulf Stream transport of about 90; about 25 x 106 m3 s-1 of this is entrained flow from a southwestward, nearly barotropic flow along the Middle Atlantic Bight continental slope.
Unlike the wind-driven, constant depth, Sverdrop transport result, total transport streamlines are closed in the western boundary without need of bottom or lateral friction in the governing equations.
Mellor, George L., and T Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems. Reviews of Geophysics & Space Physics, 20(4), 851-875. Abstract
Applications of second-moment turbulent closure hypotheses to geophysical fluid problems have developed rapidly since 1973, when genuine predictive skill in coping with the effects of stratification was demonstrated. The purpose here is to synthesize and organize material that has appeared in a number of articles and add new useful material so that a complete (and improved) description of a turbulence model from conception to application is condensed in a single article. It is hoped that this will be a useful reference to users of the model for application to either atmospheric or oceanic boundary layers.
Dickey, T D., and George L Mellor, 1980: Decaying turbulence in neutral and statified fluids. Journal of Fluid Mechanics, 99(Part 1), 13-31. Abstract
Decaying turbulence in neutral and stratified fluids has been studied experimentally for relatively high mesh Reynolds numbers and long time histories. The neutral case indicates an initial period decay law, Decaying turbulence in neutral and stratified fluids has been studied experimentally for relatively high mesh Reynolds numbers and long time histories. The neutral case indicates an initial period decay law, through non-dimensional time which is considerably longer than previous measurements at the same mesh Reynolds number (Re = 48260). The stratified experiment resulted in a decay rate virtually identical to that of the neutral case through Wgt/M = 275. However, the decay rate sharply decreased after this time when the field of turbulence was replaced by internal gravity waves. A critical Richardson number marks the transition from the turbulence to an internal gravity wave domain.
Mellor, George L., and P T Strub, 1980: Similarity solutions for the stratified turbulent Rayleigh problem. Journal of Physical Oceanography, 10(3), 455-460. Abstract PDF
The type of stratified flow suggested by the Kantha Phillips and Azad experiment is examined analytically and shown to be a self-similar, turbulent flow which includes the well-documented flatplate, turbulent boundary-layer case. Some relevant second-moment turbulent closure model calculations are compared with the KPA data.
Dickey, T D., and George L Mellor, 1979: The Kolmogoroff r2/3 law. Physics of Fluids, 22(6), 1029-1032. Abstract
The Kolmogoroff r2/3 law has been rederived from high wavenumber universal spectra and an additive constant has been obtained as a function of a the empirical constant in the r2/3law. The value of a is uncertain. However, values determined from both structure function data and spectral density data have been collected and plotted as a function of Reynolds number. Although the results may be of more general interest, the original motivation for this study was to provide a diagnostic tool for evaluating the turbulent energy dissipation rate from correlation data (much as wall shear stress is evaluated from velocity profile data using the law of the wall). The technique for this procedure is illustrated.
Yamada, T, and George L Mellor, 1979: A numerical simulation of BOMEX data using a turbulence closure model coupled with ensemble cloud relations. Quarterly Journal of the Royal Meteorological Society, 105(446), 915-944. Abstract PDF
A one-dimensional version of a simplified, second-moment turbulence closure model, coupled with a recently developed cloud model, is used to simulate BOMEX (Barbados Oceanographic and Meterological Experiment) data. Partial differential equations for the turbulence energy and a master length scale are solved. Simulated mixing ratios of water vapour, virtual temperatures and horizontal wind speeds are compared with observations. Horizontal wind speeds agree quite well; however, simulated temperature and mixing ratio of water vapour at the end of the fourth day are about 2 K and 1.5 g kg-1 higher, respoectively, than corresponding observations; possibly this is due to the fact that the surface temperature used in the simulation is too high. Mean liquid water, cloud volume, liquid water variance, turbulence energy and eddy viscosity coefficients are presented, but data for these variables are not available for comparison. Surface momentum, heat and moisture fluxes are also presented and are compared with data. Senstitivity studies indicate that the simulated mixing ratios of water vapour agree best with observations when both vertical wind and horizontal advection obtained from the data are included.
The present study is encouraging, although further research is required to improve the model and to develop confidence in its predictive capability.
Mellor, George L., 1977: The Gaussian cloud model relations. Journal of the Atmospheric Sciences, 34(2), 356-358. Abstract PDF
Sommeria and Deardorff (1977) have derived turbulence closure relations which should be important to cloud modeling. To obtain these relations they have had to invoke some analytical approximations and data from numerical statistical experiments. In the present paper, the analytical approximations have been eliminated. Somewhat surprisingly, results obtained here agree exactly with those obtained by the previous authors. Other new and useful relations are presented.
Mellor, George L., and P A Durbin, 1975: The structure and dynamics of the ocean surface mixed layer. Journal of Physical Oceanography, 5, 718-728. Abstract PDF
The present paper describes a one-dimensional unsteady model of the ocean surface mixed layer. The model somewhat resembles the approach of Munk and Anderson in that the differential equations for mean velocity and temperature are solved. The Richardson-number-dependent stability functions which enter the model are significantly different, however, as is the fact that we are able to solve problems with realistic boundary conditions. Furthermore, all empirical constants have been determined from neutral turbulent flow experiments.
Comparisons of prediction and data are favorable.
Mellor, George L., 1975: A comparative study of curved flow and density-stratified flow. Journal of the Atmospheric Sciences, 32(7), 1278-1282. Abstract PDF
A semi-empirical theory, used to predict buoyancy effects in a density- stratified and shear-driven flow, is also applied to the case of a boundary layer with curvature. Curved flow data are available and interesting in their own right since it can be seen that the Reynolds stress is reduced to zero at a critical "curvature Richardson" number predicted reasonably well by the theory.
Yamada, T, and George L Mellor, 1975: A simulation of the Wangara atmospheric boundary layer data. Journal of the Atmospheric Sciences, 32(12), 2309-2329. Abstract PDF
Previously, the authors have studied a hierarchy of turbulent boundary layer models, all based on the same closure assumptions for the triple turbulence moments. The models differ in complexity by virtue of a systematic process of neglecting certain of the tendency and diffusion terms in the dynamic equations for the turbulent moments. Based on this work a Level 3 model was selected as one which apparently sacrificed little predictive accuracy, but which afforded considerable numerical simplification relative to the more complex Level 4 model. An earlier paper had demonstrated that the model produced similarity solutions in near agreement with surface, constant flux data. In this paper, simulations from the Level 3 model are compared with two days of Wangara atmospheric boundary layer data (Clarke, et al., 1971). In this comparison, there is an easily identified error introduced by our inability to include advection of momentum in the calculation since these terms were not measured. Otherwise, the calculated results and the observational data appear to be in close agreement.
Mellor, George L., and T Yamada, 1974: A hierarchy of turbulence closure models for planetary boundary layers. Journal of the Atmospheric Sciences, 31(7), 1791-1806. Abstract PDF
Turbulence models centered on hypotheses by Rotta and Kolmogoroff are complex. In the present paper we consider systematic simplifications based on the observation that parameters governing the degree of anisotropy are small. Hopefully, we shall discern a level of complexity which is intuitively attractive and which optimizes computational speed and convenience without unduly sacrificing accuracy.
Discussion is focused on density stratified flow due to temperature. However, other dependent variables-such as water vapor and droplet density-can be treated in analogous fashion. It is, in fact, the anticipation of additional physical complexity in modeling turbulent flow fields that partially motivates the interest in an organized process of analytical simplification.
For the problem of a planetary boundary layer subject to a diurnally varying surface heat flux or surface temperature, three models of varying complexity have been integrated for 10 days. All of the models incorporate identical empirical constants obtained from neutral flow data alone. The most complex of the three models requires simultaneous solution of 10 partial differential equations for turbulence moments in addition to the equations for the mean velocity components and temperature; the least complex eliminates all of the 10 differential equations whereas a "compromise" model retains two differential equations for total turbulent energy and temperature variance.
We conclude that all of the models give nearly the same results. We find the two-differential-equation model particularly attractive.
Mellor, George L., 1973: Analytic prediction of the properties of stratified planetary surface layers. Journal of the Atmospheric Sciences, 30(6), 1061-1069. Abstract PDF
By considering the complex of one-point, turbulent moment equations for velocity, pressure and temperature, it appears possible to predict some properties of diabatic, density-stratified planetary layers using empirical information obtained from laboratory turbulence data in the absence of density stratification. In this paper, attention is focused on the near- surface, constant-flux layer. The results, like the empirical input, are simple and, hopefully, will be instructive and useful in the formulation of improved and possibly more complicated models in the future.
Mellor, George L., 1972: The large Reynolds number, asymptotic theory of turbulent boundary layers. International Journal of Engineering Science, 10, 851-873. Abstract
A self-consistent, asymptotic expansion of the one-point, mean turbulent equations of motion is obtained. Results such as the velocity defect law and the law of the wall evolve in a relatively rigorous manner, and a systematic ordering of the mean velocity boundary layer equations and their interaction with the main stream flow are obtained. The analysis is extended to the turbulent energy equation and to a treatment of the small scale equilibrium range of Kolmogoroff; in velocity correlation space the two-thirds power law is obtained. Thus, the two well-known 'laws' of turbulent flow are imbedded in an analysis which provides a great deal of other information.
Mellor, George L., 1967: Incompressible, turbulent boundary layers with arbitrary pressure gradients and divergent or convergent cross flows. American Institute of Aeronautics and Astronautics (AIAA) Journal, 5(9), 1570-1579. Abstract
An effective viscosity hypothesis that has previously led to rather detailed predictions of equilibrium turbulent boundary layers is now applied to boundary layers with arbitrary mainstream pressure variations and with divergent or convergent cross flows. The empirical content of the hypothesis involving three empirical constants (one of which is the von Karman constant) is solely derived from constant pressure profile data. The present work is similar to the previous work in that the mean differential equations of motion are integrated numerically. This time, however, one must deal with partial differential equations instead of the ordinary differential equations applicable to equilibrium flows. An important result is that prediction of the skin-friction coefficient and separation is very good. For the data considered, it is apparent that the effective viscosity hypothesis is not restricted to equilibrium flows; a corollary is that the effective viscosity can be related to the local mean velocity profile to some reasonable but undetermined degree of approximation. A further result is that for a particular three-dimensional, divergent flow experiment the measured cross- flow profiles agree with those calculated using a scalar-effective viscosity.
Mellor, George L., and D M Gibson, 1966: Equilibrium turbulent boundary layers. Journal of Fluid Mechanics, 24(2), 225-253. Abstract
Empirical information is extracted from constant-pressure flows and, on this basis alone, the equations of motion are solved for flows where the pressure gradient parameter is held constant. The experimental defect profiles of Clauser and the near-separating profile of Stratford are predicted quite well.
The present work is an extension of the work of Clauser and Townsend in that a particular form for an effective or eddy viscosity is hypothesized. Here, however, a continuous and analytically precise family of defect profiles are calculated for the entire range. The solutions span the whole profile with the exception of the viscous sublayer.
A detailed consideration of the viscous sublayer and a comparative examination of various eddy viscosity hypotheses are included in a companion paper.
