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- Author or Editor: George L. Mellor x
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Abstract
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.
Abstract
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.
Abstract
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.
Abstract
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.
Abstract
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.
Abstract
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.
Abstract
Sommeria and Deardorff (1977) have derived turbulence closure relations which should be important to cloud modeling. To obtain these relations they have hade 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.
Abstract
Sommeria and Deardorff (1977) have derived turbulence closure relations which should be important to cloud modeling. To obtain these relations they have hade 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.
Abstract
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Abstract
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 û α and, by convention, is called the “current.” The Stokes drift is labeled u Sα and the mean velocity is U α ≡ û α + u Sα . 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.
Abstract
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 û α and, by convention, is called the “current.” The Stokes drift is labeled u Sα and the mean velocity is U α ≡ û α + u Sα . 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.
Abstract
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 equation 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.
Abstract
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 equation 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.
Abstract
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 middepths (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 assimilation shows an improved skill at all depths compared to assimilation of each set of data separately.
Abstract
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 middepths (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 assimilation shows an improved skill at all depths compared to assimilation of each set of data separately.
Abstract
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 middepth (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.
Abstract
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 middepth (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.
