Nonlinear Primitive Equation Model Research Articles

Internal Tides in the Southwestern Atlantic off Brazil: Observations and Numerical Modeling

Abstract Data collected from moored instruments, deployed over the southeastern Brazilian continental shelf during the summer and winter months of 2001, show internal tide activity near the shelf break. To help to elucidate the observations, a fully three-dimensional nonlinear primitive equation model is applied to simulate the regional barotropic and baroclinic tides. Two semidiurnal (M2 and S2) and two diurnal (K1 and O1) tidal frequencies are considered. Tidal surface elevations are relatively small over the whole modeled area, reaching maximum values of about 0.40 m for M2 and 0.11 m for O1. Comparison between observed and computed tide elevation and Greenwich phase shows reasonable agreement. When the baroclinic response of the model is investigated, stratification is prescribed using summer and winter climatology data of potential density. In this case, the model response to summer and winter stratifications is very similar and internal tides are generated over the shelf break and slope, with vertical displacements up to 25 m, and seaward propagation. Modeled semidiurnal tidal ellipses agree well with winter and summer observations. Observed diurnal tidal ellipses in the middle of the continental shelf and close to the shelf break during summer show an intensification through the water column that could not be represented by the model. Estimates of the total baroclinic M2 offshore energy flux are about 3.5 and 0.5 MW considering winter and summer stratifications, respectively. Although these quantities are three orders of magnitude less than that estimated for regions known for intense internal tides, they refer to offshore fluxes computed for a very small section of the southeastern Brazilian shelf. This is the first published investigation into internal tides in the southwestern Atlantic Ocean off Brazil.

Modeling the circulation of Western Bank on the Scotian Shelf through sequential application of a variational algorithm and a nonlinear model

A new approach is presented for the modeling of the quasi‐stationary circulation through the sequential application of a variational algorithm and a nonlinear diagnostic model. The model loop begins with data assimilation of temperature, salinity, surface elevation, and velocity data in a simplified geostrophic model in which transport and continuity equations are treated as weak constraints. The temperature/salinity fields and balanced open boundary conditions points are then used as input to a nonlinear primitive equation model, which employs a turbulent closure scheme. The nonlinear model is then run to produce a diagnostic flow field. A radiation open boundary condition is applied at the outflow points of the open boundaries. These two steps are organized into an iteration cycle by using output from the nonlinear model as input to the variational model. The proposed approach combines the advantages of variational data assimilation in simplified models with a complicated fully nonlinear primitive equation model. We apply the approach to Western Bank on the Scotian Shelf. Comparisons with observed current from Western Bank, in September and October 1998, show that the sequential application of the variational approach and the fully nonlinear model allow determination of the quasi‐stationary circulation whose agreement with the observations is ≈10–30% better than circulation determined from the variational or the nonlinear model alone. Our calculations of the cross‐shelf transport across Western Bank show that it varies from 0.20 Sv to 0.35 Sv over a 2‐week period. The combined models also allow us to determine the character of the circulation over the Bank, the role of wind forcing and the implications for resident biological populations.

A regional model of the semidiurnal internal tide on the Australian North West Shelf

A fully three‐dimensional nonlinear primitive equation model is applied to a study of the semidiurnal internal tide on the Australian North West Shelf, a region of strong barotropic and baroclinic tide. The model shows a significant internal tide over most of the continental slope region, predominantly between 100‐ and 1000‐m depths. Energy flux is highly variable with regions of onshore flux and regions of offshore flux, corresponding to regions of subcritical and supercritical seabed slopes. A comparison between the model and detailed observations from one region shows reasonable agreement in vertical displacement, currents, and depth‐integrated energy flux. The comparison also highlights the difficulty in obtaining appropriate observations to provide a rigorous verification of internal tide models. The M2 and S2 internal tides show similar patterns of flow with energy in the S2 internal tide about 25% of that in the M2 constituent. Estimates of the total baroclinic M2 energy are about 2.2 GW for the North West Shelf, nearly an order of magnitude larger than suggested by earlier studies. This represents about 5% of the dissipation of the M2 barotropic tide in the model domain. The seasonal variability in the baroclinic energy flux is about 20% from the mean. The baroclinic energy extracted from the tide is expected to make a significant contribution to mixing in the region.

Three‐dimensional modeling of tidal currents and mixing quantities over the Newfoundland Shelf

Major semidiurnal (M2, S2, and N2) and diurnal (K1 and O1) barotropic tides over the Newfoundland Shelf are computed using a three‐dimensional nonlinear primitive equation model, with the vertical eddy viscosity calculated from a level 2.5 turbulence closure scheme. Computed elevation cotidal charts for the five constituents are generally consistent with previous knowledge for this region. Comparisons based on a statistical analysis of the differences between the computed elevations and currents and in situ observations indicate good agreement. While M2 tidal currents (up to 20–30 cm/s) are dominant, there are locally intensified diurnal currents (up to 5–10 cm/s) in some outer shelf locations. The diurnal current intensification is attributed to first‐mode continental shelf waves. An examination is carried out for the vertical structure of the computed M2 current and for the temporal and spatial variability of model turbulent kinetic energy, mixing length scale, vertical eddy viscosity, and bottom friction velocity. The examination indicates that large vertical eddy viscosity magnitudes and bottom friction velocities are associated with strong currents in shallow regions, where strong vertical shears produce large turbulent kinetic energy. Solutions with both specified and evolving vertical stratification indicate that the stratification has a significant influence on the vertical profile of tidal mixing parameters and currents in shallow areas. Tidally induced turbulence is substantially reduced in the bottom boundary layer and completely suppressed above it. Tidal currents decrease in the log layer, increase significantly in the rest of the bottom boundary layer, and decrease in the upper and middle water column.

Assimilation of TOPEX/Poseidon altimeter data into a global ocean circulation model: How good are the results?

The feasibility of assimilating satellite altimetry data into a global ocean general circulation model is studied. Three years of TOPEX/Poseidon data are analyzed using a global, three‐dimensional, nonlinear primitive equation model. The assimilation's success is examined by analyzing its consistency and reliability measured by formal error estimates with respect to independent measurements. Improvements in model solution are demonstrated, in particular, properties not directly measured. Comparisons are performed with sea level measured by tide gauges, subsurface temperatures and currents from moorings, and bottom pressure measurements. Model representation errors dictate what can and cannot be resolved by assimilation, and its identification is emphasized.

Observations of El Niño‐Southern Oscillation induced Rossby waves in the northeast Pacific using in situ data

Bathythermometric observations from standard National Oceanographic Data Center data sets are used to show evidence of Rossby waves in the northeast Pacific Ocean following several El Niño‐Southern Oscillation events between 1968 and 1991. Observations of a westward propagating warm anomaly following the 1982–1983 El Niño are consistent with evidence from ocean models and satellite observations that coastally trapped Kelvin waves of tropical origin excite Rossby waves in the northeast Pacific. Observed Rossby wave phase speeds are about 1.6 cm s−1 higher than those predicted by linear shallow‐water theory. At midlatitudes, observed waves are more than 1.5 times faster than theory. Phase speeds of Rossby waves in a resolution nonlinear primitive equation model exceed standard linear theory, but only by about 0.1 cm s−1, despite the fact that the model should replicate mechanisms that have recently been proposed to explain the observed anomalously fast phase speeds.

Twelve monthly experiments of 4D-variational assimilation in the tropical Atlantic during 1987: Part 2: Oceanographic interpretation

GEOSAT sea level anomalies, XBT temperature profiles and Levitus climatologies were assimilated monthly during 1987 into a non-linear primitive equation model of the tropical Atlantic ocean. Assimilating GEOSAT sea level anomalies increased the variability of the north equatorial currents when it was too weak without assimilation. The assimilated run simulates both the 1987 to 1988 sea level rise in the equatorial band and lowering in the tropics. The comparison with Reynolds satellite SST data is useful to identify the upwelling areas as the zones of highest error for the model SST. The SST warming of early 1988 is little changed by the assimilation. It confirms that this warming was determined more by the wind forcing than by the internal dynamics. As previously deduced from inverted echo sounders, we find that the North Equatorial Counter Current (NECC) increased its transport during 1987. The assimilation reveals in addition that the NECC formed a strong gyre with the North Equatorial Current (NEC). Consistently with previous analysis, this strengthening of the gyre was a result of the anomaly in ITCZ position and intensity. The heat gain by the Equatorial Under Current (EUC) was reduced because of this northward drift of the NECC. The South equatorial Current (SEC) was strong in spring and summer, moving warm surface water to the west. The eastward transport of the underlying EUC was thus reduced, leading to a deficit of warm water in the Gulf of Guinea in 1987. Comparisons of the assimilated northward heat transport (NHT) with climatological data estimates principally suggests that there was a heat gain of up to 1 petaWatt in the northern tropics (3°N–18°N) in 1987. The assimilated NHT was nil at 25°N and negative within 5°S–15°S. The 1987–88 warm events in the tropical Atlantic ocean presents a temporal progression from the north to the south. The assimilation presents an increased transport of warm water by the EUC after September 1987. It confirms the positive impact of the mass redistribution on the subsurface warming of the Gulf of Guinea. The other significant impact in 1988 is an increase of the gyre formed by the SEC and the South Equatorial Counter Current (SECC) and an inhibition of the coastal upwelling off Namibia.

Twelve monthly experiments of 4D-variational assimilation in the tropical Atlantic during 1987: Part 1: Method and statistical results

GEOSAT sea level anomalies, XBT temperature profiles and Levitus (1982) climatologies are assimilated monthly during 1987 into a non-linear primitive equation model of the tropical Atlantic ocean. The model has a resolution of 1° longitude × 1/3° latitude × 20 vertical levels extending from 50°S to 50°N. The model physics are complex, including in particular an imbedded 1D-mixed layer model and a 3D-Richardson number dependent mixing under the mixed layer. The model is forced monthly by ship winds and Oberhüber (1988) climatologies of heat and fresh water fluxes. A 2° longitude × 1° latitude gridded optimal interpolation analysis of GEOSAT sea level anomalies is assimilated between 20°S and 20°N at each grid point of the model. Validated TOGA XBT profiles are also assimilated between 30°S and 30°N at their exact location. Decorrelation in space and time is assumed for the error structures. Data weights are taken to vary with data accuracy and with a proxy function of the model representativity. The variational assimilation succeeds in determining initial conditions that reduce the model-data misfits and demonstrates that the temperature and the sea surface height of a general circulation model are sensitive to initial conditions. Despite the complex evolution of the present model over one month, the adjoint model is able to provide efficient descent directions for the minimization algorithm. Thus, 4D-variational assimilation can be used in the tropics with a general circulation model. The assimilation of satellite altimetry helps to reduce the misfit between the model and the XBT profiles. The decay of the XBT misfit is faster and the corrections are not localized to the XBT positions but influence the whole basin. The biggest impact appears where the GEOSAT variability is the highest, e.g. around the North Equatorial Counter Current and the North Equatorial Current (5°N–15°N). The assimilated sea level variability is close to the GEOSAT variability. On the other hand, the spatial variability of the modelled temperatures is less well defined in comparison with the XBT profiles; although the large scale is reproduced, details are missing in the vertical stratification at the surface and in the thermocline. The modelled equatorial upwelling is also too strong after assimilation with a 2°C cold bias at the surface in July. The 11 one-month forecasts in 1987 following each month of assimilation show an improvement of the simulation with reference to XBT profiles below the mixed layer. Temperature profiles in the mixed layer are neither improved nor degraded. This could indicate the need of a better time resolution of the forcings, for example including the diurnal cycle and a prognostic model of the mixed layer. Compared to GEOSAT, the forecasts of 1987 show an improvement up until May. The equatorial upwelling of July is degraded by the assimilation, because of a combination of the heat flux correction and the surface cold bias which is increased by the assimilation. This demonstrates that the inadequacy between the heat fluxes and the mixed layer model is a critical issue with assimilation in thermodynamical models. The pure forecast over the first 7 months of 1988 compares well to XBT profiles under the thermocline, showing that the initialization of the model has a positive impact for more than 7 months. The description of the warm anomaly of early 1988 relative to 1987 is globally improved by the assimilation. In particular, the variability of the north equatorial currents measured by altimetry is only reproduced after data assimilation. There is a benefit for the year-to-year signal in the Gulf of Guinea. This emphasizes the importance of the in situ observing network and satellite altimetric data.

Impact of tropical heat sources on the South American tropospheric upper circulation and subsidence

The nonlinear adjustment of the wind and mass fields to idealized tropical heat sources is studied with a simple nonlinear primitive equation model with emphasis on the upper level circulation over South America and neighboring oceanic regions during the austral summer. Numerical experiments are performed with (1) an idealized symmetrical heat source in the Amazon region, (2) the asymmetry induced in source (1) by the SACZ, (3) the effect of the Atlantic ITCZ off the Amazon mouth, (4) the African heat source, (5) the West Pacific source, and (6) the central Pacific source during the warm phase of ENSO. The linear response is obtained through the reduction of the heat source by a factor of 10 and subsequent multiplication of the results by the same factor. Two basic questions are discussed: (1) are localized heat sources important for the development of the observed cyclonic flow in the midequatorial Atlantic and (2) where is the compensating subsidence associated with the Amazon heat source located? The nonlinearity helps organizing a weak cyclonic curvature in the midequatorial Atlantic, with the inclusion of source (2). The basic state generated by the west Pacific source, and primarily by the central Pacific source, has a large impact on the cyclonic curvature on the equatorial Atlantic. The compensating subsidence associated with the Amazon source is concentrated on the southwest side of the source. The SACZ extension helps to enhance the subsidence over the northern Argentina, and the Atlantic ITCZ enhances the subsidence over northeast Brazil and central equatorial Atlantic. Nonlinearity weakens the subsidence at the 500 hPa level inducing a more barotropic structure in the dynamical response to the heating.

The effect of barotropic shear on baroclinic instability Part I: Normal mode problem

Abstract The effect of barotropic shear in the basic flow on baroclinic instability is investigated using a linear multilevel quasi-geostrophic β-plane channel model and a nonlinear spherical primitive equation model. Barotropic shear has a profound effect on baroclinic instability. It reduces the growth rates of normal modes by severely restricting their structure, confirming earlier results with a two-layer model. Dissipation, in the form of Ekman pumping and Newtonian cooling, does not change the main characteristics of the effect of the shear on normal mode instability. Barotropic shear in the basic state, characterized by large shear vorticity with small horizontal curvature, also effects the nonlinear development of baroclinic waves. The shear limits the energy conversion from the zonal available potential energy to eddy energy, reducing the maximum eddy kinetic energy level reached by baroclinic waves. Barotropic shear, which controls the level of eddy activity, is a major factor which should be considered when parameterizing the eddy temperature and momentum fluxes induced by baroclinic waves in a climate model.

An approximate Kaiman filter for ocean data assimilation: An example with an idealized Gulf Stream model

A practical method of data assimilation for use with large, nonlinear, ocean general circulation models is explored. A Kaiman filter based on approximations of the state error covariance matrix is presented, employing a reduction of the effective model dimension, the error's asymptotic steady state limit, and a time‐invariant linearization of the dynamic model for the error integration. The approximations lead to dramatic computational savings in applying estimation theory to large complex systems. We examine the utility of the approximate filter in assimilating different measurement types using a twin experiment of an idealized Gulf Stream. A nonlinear primitive equation model of an unstable east‐west jet is studied with a state dimension exceeding 170,000 elements. Assimilation of various pseudomeasurements are examined, including velocity, density, and volume transport at localized arrays and realistic distributions of satellite altimetry and acoustic tomography observations. Results are compared in terms of their effects on the accuracies of the estimation. The approximate filter is shown to outperform an empirical nudging scheme used in a previous study. The examples demonstrate that useful approximate estimation errors can be computed in a practical manner for general circulation models.

A simple three‐dimensional model of the circulation off Newfoundland

A simple numerical model is developed to study the three‐dimensional structure of the steady circulation off Newfoundland. The model uses linear dynamics, a vertical eddy viscosity to parameterize the vertical mixing, and a linear parameterization of bottom friction in which the bottom stress is related to the bottom velocity. Realistic geometry and bottom topography are included. The results presented here are for an ocean of uniform density. The model is driven by specifying the sea level along the northern boundary, the inflow corresponding to the inshore and offshore branches of the Labrador Current. The model can reproduce the circulation pattern found by Greenberg and Petrie (1988), who used a vertically integrated nonlinear primitive equation model. It also enables dynamical interpretation to be given in terms of flow along f/h contours and frictional “diffusion” across these contours. In this connection, we investigate the dependence on the friction parameters of the transport through several sections. We also examine the onshore‐offshore mass exchange over the southeastern edge of the Grand Banks. We find that the model Labrador Current flowing southward along the shelf break leads to an onshelf flux in the upper part of the water column and an offshelf flux in the bottom Ekman layer. This is caused by the frictional torque and leads to an along‐slope decay of the current amplitude. Finally the limitations of the model and plans for its further development are discussed.

Baroclinic Instability in Vertically Discrete Systems

Two vertically discrete systems, one based on the “Charney-Phillips grid” and the other on the “Lorenz grid,” are compared in view of the quasi-geostrophic potential vorticity equation and baroclinic instability. It is shown that with the Charney-Phillips grid, the standard grid for the quasi-geostrophic system of equations, one can easily maintain important dynamical constraints on quasi-geostrophic flow, such as the conservation of quasi-geostrophic potential vorticity through horizontal advection and resulting integral constraints. With the Lorenz grid, however, in which horizontal velocity and (potential) temperature are carried at same levels, it is not straightforward even to define quasi-geostrophic potential vorticity. Moreover, due to an extra degree of freedom in potential temperature, the Lorenz grid can falsely satisfy the necessary condition for baroclinic instability near the lower and upper boundaries. In fact, eigenvalue solutions of the linear quasi-geostrophic equations show the existence of spuriously amplifying modes with short wavelengths, one trapped near the lower boundary and the other near the upper boundary. The former grows more rapidly then the latter when static stability increases with height. In a model discretized both in vertical and horizontal, the spurious amplification appears with high horizontal resolution unless vertical resolution is very high. The existence of the spurious amplification of short waves in a nonlinear primitive equation model is also confirmed. Here the amplification also influences longer waves though nonlinearity and upper level presumably through vertical propagation of gravity waves. It is shown that the spurious amplification can be removed at its origin by introducing additional terms in the thermodynamic equations for the bottom and top layers, which effectively eliminate the possibility of falsely satisfying the necessary condition for baroclinic instability.

Nonlinear Instability and Error Growth in Northern Hemisphere Three-Dimensional Flows: Cyclogenesis, Onset-of-Blocking and Mature Anomalies

Abstract The nonlinear interactions of an initial small amplitude instability mode with a three-dimensional climatological flow (for Northern Hemisphere, January 1978), an which it is growing are studied in a multilevel nonlinear primitive equation model incorporating spherical geometry. Numerical experiments in which there is either constant or no forcing are considered. For the forced runs the forcing is such that the climatological flow is an exact steady state solution which, however, is unstable when perturbed by the normal mode disturbance. For the unforced runs, we examine the divergence of two “forecasts,” one of which has the climatological flow as initial conditions and the other, the climatological flow perturbed by the instability solution. The structural changes that occur in the total and disturbance streamfunctions and kinetic energies and in the disturbance momentum and heat fluxes are examined. The initially small-scale shallow disturbance increases in wale and penetrates into the upper t...

Numerical Simulation of Nonlinear Jet Streak Adjustment

Abstract The geostrophic adjustment process in a propagating jet maximum is studied through numerical experiments performed with a two-layer, nonlinear primitive equation model. Gravity-inertia modes generated by the jet streak from an initial three-dimensional balanced flow are isolated with the aid of diagnostic quasi-geostrophic and balance models. Them modes appear to represent gravitational subsynoptic “signal” rather than “noise,” and their behavior is examined as a function of initial core strength. Persistent unbalanced patterns relative to the jet core are discovered, whose amplitude increases approximately as the square of the Rossby number. Their behavior is suggestive of forced, rather than free, ageostrophic modes, and their existence implies that the “four-cell” vertical motion pattern characteristic of jet streaks in filtered models may be significantly altered in a real jet streak. The free response to initial data imbalances placed at varying locations in a jet streak environment is explo...

The Downstream and Upstream Development of Unstable Baroclinic Waves

The response of a baroclinically unstable atmosphere to a localized initial perturbation is examined. In a preliminary numerical experiment, using a nonlinear primitive-equation model with spherical geometry, new disturbances grow regularly downstream in broad agreement with a particular type of observed development. The initial disturbance moves downstream, while smaller scale disturbances develop upstream of it—each forming at about the same longitude. Later downstream disturbances have upper level amplitudes significantly larger than found in nonlinear integrations using normal-mode initial conditions. Similar development is found in a quasi-geostrophic model with spherical geometry, and the rate of spreading of the instability is insensitive to the amplitude of the initial perturbation. For the baroclinic instability model considered by Eady the downstream disturbances appear first at the upper surface at a position which moves with a speed close to that of the zonal-mean flow at this surface, while the upstream disturbances all form close to a fixed location it the surface mean flow is zero. New disturbances possess a short horizontal length scale which subsequently expands toward that of the most unstable normal mode, and their amplitude increases at a rate initially much in excess of normal-mode growth rates. Phase speed is close to that of normal modes. Diagnosis is performed both using a form of the omega equation and through an asymptotic analysis of the solution represented by a sum of normal modes of different wavenumbers. Further calculations have been performed for a linearized β-plant model. Inclusion of a mean density variation with height and the β-effect slows the downstream spread of the instability by up to 15%, and this downstream development occurs on a longer length scale. The upstream disturbances quite generally develop close to a fixed location when the mean surface flow is zero, and possess a relatively short zonal scale if the low-level mean flow has vertical shear. An additional primitive-equation integration confirms the results of the preliminary nonlinear experiment.