Two-layer Primitive Equation Model Research Articles

A Four-Dimensional Incremental Analysis Update for the Ensemble Kalman Filter

Abstract The analysis produced by the ensemble Kalman filter (EnKF) may be dynamically inconsistent and contain unbalanced gravity waves that are absent in the real atmosphere. These imbalances can be exacerbated by covariance localization and inflation. One strategy to combat the imbalance in the analyses is the incremental analysis update (IAU), which uses the dynamic model to distribute the analyses increments over a time window. The IAU has been widely used in atmospheric and oceanic applications. However, the analysis increment that is gradually introduced during a model integration is often computed once and assumed to be constant for an assimilation window, which can be seen as a three-dimensional IAU (3DIAU). Thus, the propagation of the analysis increment in the assimilation window is neglected, yet this propagation may be important, especially for moving weather systems. To take into account the propagation of the analysis increment during an assimilation window, a four-dimensional IAU (4DIAU) used with the EnKF is presented. It constructs time-varying analysis increments by applying all observations in an assimilation window to state variables at different times during the assimilation window. It then gradually applies these time-varying analysis increments through the assimilation window. Results from a dry two-layer primitive equation model and the NCEP GFS show that EnKF with 4DIAU (EnKF-4DIAU) and 3DIAU (EnKF-3DIAU) reduce imbalances in the analysis compared to EnKF without initialization (EnKF-RAW). EnKF-4DIAU retains the time-varying information in the analysis increments better than EnKF-3DIAU, and produces better analysis and forecast than either EnKF-RAW or EnKF-3DIAU.

Adaptive Localization for the Ensemble-Based Observation Impact Estimate Using Regression Confidence Factors

Abstract The goal of this study is to improve an ensemble-based estimation for forecast sensitivity to observations that is straightforward to apply using existing products of any ensemble data assimilation system. Because of limited ensemble sizes compared to the large degrees of freedom in typical models, it is necessary to apply localization techniques to obtain accurate estimates. Fixed localization techniques do not guarantee accurate impact estimates, because as forecast time increases the error correlation structures evolve with the flow. Here a dynamical localization method is applied to improve the observation impact estimate. The authors employ a Monte Carlo “group filter” technique to limit the effects of sampling error via regression confidence factor (RCF). Experiments make use of the local ensemble transform Kalman filter (LETKF) with a simple two-layer primitive equation model and simulated observations. Results show that the shape, location, time dependency, and variable dependency of RCF localization functions are consistent with underlying dynamical processes of the model. Application of RCF localization to ensemble-estimated impact showed marked improvement especially for longer forecasts and at midlatitudes, when systematically verified against actual impact in RMSE and skill scores. The impact estimates near the equator were not as effective because of large discrepancies between the RCF function and the localization used at assimilation time. These latter results indicate that there exists an inherent relationship between the localization applied during the assimilation time and the proper localization choice for observation impact estimates. Application of RCF for automatically tuned localization is introduced and tested for a single observation experiment.

Wind- versus Eddy-Forced Regional Sea Level Trends and Variability in the North Pacific Ocean

Abstract Regional sea level trend and variability in the Pacific Ocean have often been considered to be induced by low-frequency surface wind changes. This study demonstrates that significant sea level trend and variability can also be generated by eddy momentum flux forcing due to time-varying instability of the background oceanic circulation. Compared to the broad gyre-scale wind-forced variability, the eddy-forced sea level changes tend to have subgyre scales and, in the North Pacific Ocean, they are largely confined to the Kuroshio Extension region (30°–40°N, 140°–175°E) and the Subtropical Countercurrent (STCC) region (18°–28°N, 130°–175°E). Using a two-layer primitive equation model driven by the ECMWF wind stress data and the eddy momentum fluxes specified by the AVISO sea surface height anomaly data, the relative importance of the wind- and eddy-forced regional sea level trends in the past two decades is quantified. It is found that the increasing (decreasing) trend south (north) of the Kuroshio Extension is due to strengthening of the regional eddy forcing over the past two decades. On the other hand, the decreasing (increasing) sea level trend south (north) of the STCC is caused by the decadal weakening of the regional eddy momentum flux forcing. These decadal eddy momentum flux changes are caused by the background Kuroshio Extension and STCC changes in connection with the Pacific decadal oscillation (PDO) wind pattern shifting from a positive to a negative phase over the past two decades.

A two-layer primitive equation model of an idealized Strait of Gibraltar connected to an eastern basin

We describe the solutions of a numerical two-layer primitive equation model of an idealized Strait of Gibraltar and an adjacent eastward basin. The quasi-steady circulation pattern in the basin features an anticyclonic gyre southward of the strait exit and an eastward boundary current attached to the southern boundary of the basin. A variation of the initial upper-layer depth and the target interface height at the eastern boundary causes minor changes of the upper-layer circulation in a basin with rectangular coastline and larger changes when a cape is added to the coastline of the southern boundary.

On Imbalance Generated by Vortical Flows in a Two-Layer Spherical Boussinesq Primitive Equation Model

Abstract The spontaneous adjustment emission of inertia–gravity waves is investigated by examining the amount of imbalance generated during the evolution of unstable jets in an isentropic two-layer primitive equation model on the sphere. To determine the balance and thus the imbalance, potential vorticity (PV) inversion by means of the first-, second-, and third-order plain-δδ, plain-δγ, and plain-γγ balance relations, as well as the Bolin–Charney balance relations, is applied. Five sets of experiments are carried out by (i) changing the upper-layer potential temperature with a fixed initial jet speed and (ii) changing the initial jet speed with a fixed upper-layer potential temperature. For each set, the relation between the optimal measure of imbalance with the corresponding measure of balanced vortical flow is sought at the peak of instability. The minimal imbalance obtained by 1 of the above 10 PV inversion procedures is considered as the optimal. The focus herein is on the magnitude of imbalance and its scaling with various measures of balanced flow. It is shown that for the magnitude of imbalance it is always possible to find the proper measure with an exponentially small scaling. For the imbalance-to-balance ratio, however, a power law is obtained when either the jet or the stratification is weak. The latter difference in scaling behavior of the imbalance and imbalance-to-balance ratio may be an artifact of the inevitable numerical errors whose effects are felt more strongly when the signature of vortical flow is weak.

A Comparison of the Hybrid and EnSRF Analysis Schemes in the Presence of Model Errors due to Unresolved Scales

Abstract A hybrid analysis scheme is compared with an ensemble square root filter (EnSRF) analysis scheme in the presence of model errors as a follow-up to a previous perfect-model comparison. In the hybrid scheme, the ensemble perturbations are updated by the ensemble transform Kalman filter (ETKF) and the ensemble mean is updated with a hybrid ensemble and static background-error covariance. The experiments were conducted with a two-layer primitive equation model. The true state was a T127 simulation. Data assimilation experiments were conducted at T31 resolution (3168 complex spectral coefficients), assimilating imperfect observations drawn from the T127 nature run. By design, the magnitude of the truncation error was large, which provided a test on the ability of both schemes to deal with model error. Additive noise was used to parameterize model errors in the background ensemble for both schemes. In the first set of experiments, additive noise was drawn from a large inventory of historical forecast errors; in the second set of experiments, additive noise was drawn from a large inventory of differences between forecasts and analyses. The static covariance was computed correspondingly from the two inventories. The hybrid analysis was statistically significantly more accurate than the EnSRF analysis. The improvement of the hybrid over the EnSRF was smaller when differences of forecasts and analyses were used to form the random noise and the static covariance. The EnSRF analysis was more sensitive to the size of the ensemble than the hybrid. A series of tests was conducted to understand why the EnSRF performed worse than the hybrid. It was shown that the inferior performance of the EnSRF was likely due to the sampling error in the estimation of the model-error covariance in the mean update and the less-balanced EnSRF initial conditions resulting from the extra localizations used in the EnSRF.

A Comparison of Hybrid Ensemble Transform Kalman Filter–Optimum Interpolation and Ensemble Square Root Filter Analysis Schemes

Abstract A hybrid ensemble transform Kalman filter (ETKF)–optimum interpolation (OI) analysis scheme is described and compared with an ensemble square root filter (EnSRF) analysis scheme. A two-layer primitive equation model was used under perfect-model assumptions. A simplified observation network was used, and the OI method utilized a static background error covariance constructed from a large inventory of historical forecast errors. The hybrid scheme updated the ensemble mean using a hybridized ensemble and static background-error covariance. The ensemble perturbations in the hybrid scheme were updated by the ETKF scheme. The EnSRF ran parallel data assimilation cycles for each member and serially assimilated the observations. The EnSRF background-error covariance was estimated fully from the ensemble. For 50-member ensembles, the analyses from the hybrid scheme were as accurate or nearly as accurate as those from the EnSRF, depending on the norm. For 20-member ensembles, the analyses from the hybrid scheme were more accurate than analyses from the EnSRF under certain norms. Both hybrid and EnSRF analyses were more accurate than the analyses from the OI. Further reducing the ensemble size to five members, the EnSRF exhibited filter divergence, whereas the analyses from the hybrid scheme were still better than those updated by the OI. Additionally, the hybrid scheme was less prone to spurious gravity wave activity than the EnSRF, especially when the ensemble size was small. Maximal growth in the ETKF ensemble perturbation space exceeded that in the EnSRF ensemble perturbation space. The relationship of the ETKF ensemble variance to the analysis error variance, a measure of a spread–skill relationship, was similar to that of the EnSRF ensemble. The hybrid scheme can be implemented in a reasonably straightforward manner in the operational variational frameworks, and the computational cost of the hybrid is expected to be much less than the EnSRF in the operational settings.

Accounting for the Error due to Unresolved Scales in Ensemble Data Assimilation: A Comparison of Different Approaches

Abstract Insufficient model resolution is one source of model error in numerical weather predictions. Methods for parameterizing this error in ensemble data assimilations are explored here. Experiments were conducted with a two-layer primitive equation model, where the assumed true state was a T127 forecast simulation. Ensemble data assimilations were performed with the same model at T31 resolution, assimilating imperfect observations drawn from the T127 forecast. By design, the magnitude of errors due to model truncation was much larger than the error growth due to initial condition uncertainty, making this a stringent test of the ability of an ensemble-based data assimilation to deal with model error. Two general methods, “covariance inflation” and “additive error,” were considered for parameterizing the model error at the resolved scales (T31 and larger) due to interaction with the unresolved scales (T32 to T127). Covariance inflation expanded the background forecast members’ deviations about the ensemble mean, while additive error added specially structured noise to each ensemble member forecast before the update step. The method of parameterizing this model error had a substantial effect on the accuracy of the ensemble data assimilation. Covariance inflation produced ensembles with analysis errors that were no lower than the analysis errors from three-dimensional variational (3D-Var) assimilation, and for the method to avoid filter divergence, the assimilations had to be periodically reseeded. Covariance inflation uniformly expanded the model spread; however, the actual growth of model errors depended on the dynamics, growing proportionally more in the midlatitudes. The inappropriately uniform inflation progressively degradated the capacity of the ensemble to span the actual forecast error. The most accurate model-error parameterization was an additive model-error parameterization, which reduced the error difference between 3D-Var and a near-perfect assimilation system by ∼40%. In the lowest-error simulations, additive errors were parameterized using samples of model error from a time series of differences between T63 and T31 forecasts. Scaled samples of differences between model forecast states separated by 24 h were also tested as additive error parameterizations, as well as scaled samples of the T31 model state’s anomaly from the T31 model climatology. The latter two methods produced analyses that were progressively less accurate. The decrease in accuracy was likely due to their inappropriately long spatial correlation length scales.

Linear instabilities of a two-layer geostrophic surface front near a wall

The development of linear instabilities on a geostrophic surface front in a two-layer primitive equation model on an f-plane is studied analytically and numerically using a highly accurate differential shooting method. The basic state is composed of an upper layer in which the mean flow has a constant potential vorticity, and a quiescent lower layer that outcrops between a vertical wall and the surface front (defined as the line of intersection between the interface that separates the two layers and the ocean’s surface). The characteristics of the linear instabilities found in the present work confirm earlier results regarding the strong dependence of the growth rate ( i) on the depth ratio r (defined as the ratio between the total ocean depth and the upper layer’s depth at infinity) for r 2 and their weak dependence on the distance L between the surface front and the wall. These earlier results of the large r limit were obtained using a much coarser, algebraic, method and had a single maximum of the growth rate curve at some large wavenumber k. Our new results, in the narrow range of 1.005 r 1.05, demonstrate that the growth rate curve displays a second lobe with a local (secondary) maximum at a nondimensional wavenumber (with the length scale given by the internal radius of deformation) of 1.05. A new “fitting function” 0.183 r 0.87 is found for the growth rate of the most unstable wave ( imax) for r ranging between 1.001 and 20, and for L 2 Rd (i.e. where the effect of the wall becomes negligible). Therefore, imax converges to a finite value for r 1 1 (infinitely thin lower layer). This result differs from quasi-geostrophic, analytic solutions that obtain for the no wall case since the QG approximation is not valid for very thin layers. In addition, an analytical solution is derived for the lower-layer solutions in the region between the wall and the surface front where the upper layer is not present. The weak dependence of the growth rate on L that emerges from the numerical solution of the eigenvalue problem is substantiated analytically by the way L appears in the boundary conditions at the surface front. Applications of these results for internal radii of deformation of 35–45 km show reasonable agreement with observed meander characteristics of the Gulf Stream downstream of Cape Hatteras. Wavelengths and phase speeds of (180–212 km, 39–51 km/day) in the vicinity of Cape Hatteras were also found to match with the predicted dispersion relationships for the depth-ratio range of 1 r 2. 1. Division of Meteorology and Physical Oceanography, Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida 33149, U.S.A. 2. Corresponding author. email: ahaza@rsmas.miami.edu 3. Department of Atmospheric Sciences, Institute of Earth Sciences, Hebrew University of Jerusalem, Jerusalem 91904, Israel. Journal of Marine Research, 62, 639–662, 2004

On the Influence of Bottom Topography on the Agulhas Eddy

Abstract A series of numerical experiments with a two-layer primitive equation model is presented to study the dynamics of Agulhas eddies. The main goal of the paper is to examine the influence of an underwater meridional ridge (modeled after the Walvis Ridge) on an Agulhas eddy hitting it. First, the propagation of an eddy of the specified vertical structure over a flat bottom is considered, varying the initial eddy horizontal scale from 40 to 120 km. Unlike small nonlinear eddies, large nonlinear eddies (on the scale of Agulhas eddies) do not rapidly evolve into a compensated state (no motion in the lower layer). Second, the influence of a ridge on eddies of differing vertical structures having a specified intensity in the upper layer and a prescribed horizontal scale is analyzed. Significantly baroclinic eddies can cross the Walvis Ridge, but barotropic or near-barotropic ones cannot. The evolution of eddies crossing the ridge is compared with that of initially identical eddies moving over a flat botto...

Baroclinic Instability in a Two-Layer Model with Parameterized Slantwise Convection

Abstract A two-layer primitive equation model with a parameterization for slantwise convection is used to study explosive cyclones. The parameterization includes a simple representation of the planetary boundary layer, as well as deep and shallow cloud types. The major results are 1) the two-layer model produces a realistic explosive cyclone, 2) the explosive deepening coincides with the formation of the “bent-back warm front,” 3) fronto-genetic calculations reveal an indirect circulation due to “tilting” in the intense warm front, and 4) the quasi-Lagrangian heat and vorticity budgets indicate strong interaction of the lower- and upper-level flow during the rapid spinup stage. These results suggest a sequence of events involved in some explosive cyclogenesis: Convection leads to rapid frontogenesis and the formation of a bent-back warm front. The sudden surge of cold advection in the regions of the bent-back warm front then forces the upper-level heights over the cyclone center to fall in a rather dramat...

Assimilation of Altimeter Data in a Two-Layer Primitive Equation Model of the Gulf Stream

A two-layer finite depth, primitive equation model of the Gulf Stream region is used to study the effect of updating the model with simulated altimeter data as observations. In this study both a complete field of sea surface height (SSH) and SSH sampled along satellite tracks are used as “observations.” A simulated 17-day repeat orbit corresponding to the Geosat-ERM and a 10-day repeat period corresponding to Topex/Poseidon are used. Satellite observations give only information about the sea surface height but previous studies have shown that it is important to transfer the surface information to the lower layer as fast as possible in order for the model to have a realistic evolution. A statistical inference technique is therefore used to update the lower-layer pressure field. The velocity fields in both layers are updated using a geostrophic correction calculated from the change in the pressure fields. It is shown that updating the velocities is important for the assimilation to be successful. When complete fields of sea surface heights from an identical twin experiment are assimilated the rms error between the model solution and the “true” ocean for the upper-layer pressure field is reduced to less than 5% after six weeks of assimilation. Assimilation of simulated observations along Geosat-ERM and Topex/Poseidon tracks show that both satellites give similar levels of rms. error at the end of the assimilation period. The effect of assimilating altimeter data from two satellites is also discussed. The results show there can be a reduction in the rms error of up to 40% with the addition of a second satellite with appropriate orbital characteristics.

Sensitivity and realism of wind-driven tropical ocean models

The degree of realism of several equatorial ocean models is investigated by comparing modeled and observed mean seasonal cycles in the tropical Atlantic for two variables that play a role in sea surface temperature dynamics: 0/400 db dynamic height and surface current. The model-reality intercomparisons are made using a multivariate statistical testing procedure that takes into account the uncertainties of the atmospheric and oceanic data. In particular, the uncertainties in the model response due to random errors in the wind forcing and its interannual variability are considered and the uncertainty in the bulk formulation for the wind stress is represented. Two simplified equatorial ocean models are considered: the linear multimode model of Cane (1984) and the two-layer primitive equation model of Andrich (1989). Some results are also given for the general circulation model of Philander and Pacanowski (1986), although the effect of forcing uncertainties could not be taken into account. It is found that the errors that were considered are unable to explain the large discrepancies between model simulations and observations. In general, models represent better the 0/400 db dynamic height than the surface currents and the seasonal variations than the yearly mean. There is no striking difference in the performance of the three models for the dynamic height, but the general circulation model is superior for the surface currents, in particular near the western boundary. Better parameter tuning should, however, improve the performances of the two-layer model. It is also found that the sensitivity to forcing uncertainties is very model-dependent.

A Numerical Study of Loop Current Eddy Interaction with Topography in the Western Gulf of Mexico

Abstract Anticyclones originating from the Loop Current are known to propagate into the western Gulf of Mexico. Their frequency of generation, their long lifetimes, and satellite data suggest that at any one time one or more eddies may occupy the Gulf. Given the eddy sizes (100–200 km) and geometric confinement of the Gulf, it would appear that there may be significant interactions of individual eddies and/or interactions of the eddies with bottom topography. These possibilities are explored through the use of a two-layer primitive equation model. Variable parameters in this model study are eddy strength, vertical structure, lateral friction, and initial location relative to topography. Results indicate that eddy motion is governed by two dynamical regimes depending on its lower layer rotational strength. Anticyclones with significant lower layer anticyclonic structure develop offshore directed self advective tendencies associated with topographic dispersion which induces asymmetry in the eddy. Eddies wit...

The Linear Response of a Hemispheric Two–Level Primitive Equation Model to Forcing by Topography

Abstract A hemispheric solution is obtained for the topographically forced stationary linear perturbations in a two-layer primitive equation model of the atmosphere. Results are discussed for January conditions. Additionally, the linear theory of stationary perturbations in a quasi-geostrophic two-layer model with β-plane approximation is presented which shows that three types of standing waves may be excited by the topography. The structure of these waves and the conditions under which they appear are discussed. Furthermore, the influence of the surface friction and the vertical shear stress on these waves is studied. This quasi-geostrophic theory is applied to the January case. It turns out that the topography induces so-called ultralong waves without a horizontal node in the frictionless model atmosphere. Cold troughs develop at both levels over the major mountain chains such as the Himalayas and the Rocky Mountains. The standing waves in northern latitudes seem to be forced by the orography in middle ...