If model parameterizations of unresolved physics, such as the variety of upper ocean mixing processes, are to hold over the large range of time and space scales of importance to climate, they must be strongly physically based. Observations, theories, and models of oceanic vertical mixing are surveyed. Two distinct regimes are identified: ocean mixing in the boundary layer near the surface under a variety of surface forcing conditions (stabilizing, destabilizing, and wind driven), and mixing in the ocean interior due to internal waves, shear instability, and double diffusion (arising from the different molecular diffusion rates of heat and salt). Mixing schemes commonly applied to the upper ocean are shown not to contain some potentially important boundary layer physics. Therefore a new parameterization of oceanic boundary layer mixing is developed to accommodate some of this physics. It includes a scheme for determining the boundary layer depth h, where the turbulent contribution to the vertical shear of a bulk Richardson number is parameterized. Expressions for diffusivity and nonlocal transport throughout the boundary layer are given. The diffusivity is formulated to agree with similarity theory of turbulence in the surface layer and is subject to the conditions that both it and its vertical gradient match the interior values at h. This nonlocal “K profile parameterization” (KPP) is then verified and compared to alternatives, including its atmospheric counterparts. Its most important feature is shown to be the capability of the boundary layer to penetrate well into a stable thermocline in both convective and wind-driven situations. The diffusivities of the aforementioned three interior mixing processes are modeled as constants, functions of a gradient Richardson number (a measure of the relative importance of stratification to destabilizing shear), and functions of the double-diffusion density ratio, Rρ. Oceanic simulations of convective penetration, wind deepening, and diurnal cycling are used to determine appropriate values for various model parameters as weak functions of vertical resolution. Annual cycle simulations at ocean weather station Papa for 1961 and 1969–1974 are used to test the complete suite of parameterizations. Model and observed temperatures at all depths are shown to agree very well into September, after which systematic advective cooling in the ocean produces expected differences. It is argued that this cooling and a steady salt advection into the model are needed to balance the net annual surface heating and freshwater input. With these advections, good multiyear simulations of temperature and salinity can be achieved. These results and KPP simulations of the diurnal cycle at the Long-Term Upper Ocean Study (LOTUS) site are compared with the results of other models. It is demonstrated that the KPP model exchanges properties between the mixed layer and thermocline in a manner consistent with observations, and at least as well or better than alternatives.

Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization

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Statistical Analysis in Climate Research

Procedures that may be used to evaluate the operational performance of a wide spectrum of geophysical models are introduced. Primarily using a complementary set of difference measures, both model accuracy and precision can be meaningfully estimated, regardless of whether the model predictions are manifested as scalars, directions, or vectors. It is additionally suggested that the reliability of the accuracy and precision measures can be determined from bootstrap estimates of confidence and significance. Recommended procedures are illustrated with a comparative evaluation of two models that estimate wind velocity over the South Atlantic Bight.

/pdf/statistics-for-the-evaluation-and-comparison-of-models-4ytufgyhdk.pdf

Statistics for the evaluation and comparison of models

Recent research has pointed to a number of inherent disadvantages of unrotated principal components and empirical orthogonal functions when these techniques are used to depict individual modes of variation of data matrices in exploratory analyses. The various pitfalls are outlined and illustrated with an alternative method introduced to minimize these problems via available linear transformations known as simple structure rotations. The rationale and theory behind simple structure rotation and Procrustes target rotation is examined in the context of meteorological/climatological applications. This includes a discussion of the six unique ways to decompose a rotated data set in order to maximize the physical interpretability of the rotated results.



The various analytic simple structure rotations available are compared by a Monte Carlo simulation, which is a modification of a similar technique developed by Tucker (1983), revealing that the DAPPFR and Promax k = 2 rotations are the most accurate in recovering the input structure of the modes of variation over a wide range of conditions. Additionally, these results allow the investigator the opportunity to check the accuracy of the unrotated or rotated solution for specific types of data. This is important because, in the past, the decision of whether or not to apply a specific rotation has been a ‘blind decision’. In response to this, a methodology is presented herein by which the researcher can assess the degree of simple structure embedded within any meteorological data set and then apply known information about the data to the Monte Carlo results to optimize the likelihood of achieving physically meaningful results from a principal component analysis.

Rotation of principal components

Development is described of a Comprehensive Ocean-Atmosphere Data Set (COADS)—the result of a cooperative project to collect global weather observations taken near the ocean's surface since 1854, primarily from merchant ships, into a compact and easily used data set. As background, a historical overview is given of how archiving of these marine data has evolved from 1854, when systematic recording of shipboard meteorological and oceanographic observations was first established as an international activity. Input data sets used for COADS are described, as well as the processing steps used to pack input data into compact binary formats and to apply quality controls for identification of suspect weather elements and duplicate marine reports. Seventy-million unique marine reports for 1854–1979 were output from initial processing. Further processing is described, which created statistical summaries for each month of each year of the period, using 2° latitude × 2° longitude boxes. Monthly summary products are a...

A Comprehensive Ocean-Atmosphere Data Set

The Secchi disk is a circular white disk that is lowered into a natural body of water by a human observer until it disappears from view. The depth of disappearance is a visual measure of the clarity of the water. This review examines the physical and physiological basis of the Secchi disk procedure. The theory of the white disk is detailed to show the underlying assumptions and the consequent strengths and limitations of the procedure. The theory shows how to use a calibrated Secchi disk to predict illuminance levels as a function of depth. In particular it is shown how to predict the euphotic depth of a medium. Ten laws of the Secchi disk are stated verbally and in mathematical form. The laws show how variations in properties of the disk and the surrounding light field affect the depth of disappearance of the disk. Theory and examples lead to the following three main conclusions of this paper: (i) the Secchi disk reading zs,, (in meters) yields a quantitative estimate of a single apparent optical property (CX + K) (in meter-') of a natural hydrosol, where (Y is the (photopic) beam attenuation coefficient and K the (photopic) diffuse attenuation coefficient of the medium; (ii) the primary function of a Secchi disk is to provide a simple visual index of water clarity via z,, or a! + K, (iii) to extend the use of the Secchi disk by auxiliary objective electronic measurements of a or of K, or both, is to risk obviating or abusing this primary function.

https://aslopubs.onlinelibrary.wiley.com/doi/pdf/10.4319/lo.1986.31.5.0909

Secchi disk science: Visual optics of natural waters1

A technique is presented for selection of principal components for which the geophysical signal is greater than the level of noise. The level of noise is simulated by repeated sampling of principal components computed from a spatially and temporally uncorrected random process. By contrasting the application of principal components based upon the covariance matrix and correlation matrix for a given data set of cyclone frequencies, it is shown that the former is more suitable to fitting data and locating the individual variables that represent large variance in the record, while the latter is more suitable for resolving spatial oscillations such as the movement of primary storm tracks.

A Significance Test for Principal Components Applied to a Cyclone Climatology

When a numerical model's representation of a physical field is to be compared with a corresponding real observed field, it is usually the case that the numbers of realizations of model and observed field are relatively small, so that the natural procedure of producing histograms of pertinent statistics of the fields (e.g., means, variances) from the data sets themselves cannot be usually carried out. Also, it is not always safe to adopt assumptions of normality and independence of the data values. This prevents the confident use of classical statistical methods to make significance statements about the success or failure of the model's replication of the data. Here we suggest two techniques of determinable statistical power, in which small samples of spatially extensive physical fields can be made to blossom into workably large samples on which significance decisions can be based. We also introduce some new measures of location, spread and shape of multivariate data sets which may be used in conj...

Numerical Model-Reality Intercomparison Tests Using Small-Sample Statistics

Direct and inverse irradiance models in hydrologic optics1

In this paper we develop a new method for solving the transfer of radiation within a laterally finite optical medium. A new radiative transfer equation, based on a multimode approach, is developed which includes the explicit effects of the sides of the medium. This equation, derived for a box shaped medium, is exactly analogous to the plane parallel radiative transfer equation with a source term. Accordingly, the new equation is solved using the familiar plane-parallel techniques based on invariant imbedding relationships in the form of doubling and adding. The additional terms in the newly derived radiative transfer equation can be interpreted as apparent source and sink terms which arise from the lateral finiteness of the medium. The geometric and physical aspects of these source-sink terms and their influence on the solutions are discussed. Results also show that the multimode solutions compare well with the Monte Carlo simulations.

Rudolph W. Preisendorfer

Papers

Secchi disk science: Visual optics of natural waters1

A Significance Test for Principal Components Applied to a Cyclone Climatology

Numerical Model-Reality Intercomparison Tests Using Small-Sample Statistics

Direct and inverse irradiance models in hydrologic optics1

Multimode Radiative Transfer in Finite Optical Media. I: Fundamentals