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

A remote sensing surface energy balance algorithm for land (SEBAL)-1. Formulation

The behaviour of spectra and cospectra of turbulence in the surface layer is described within the framework of similarity theory using wind and temperature fluctuation data obtained in the 1968 AFCRL Kansas experiments. With appropriate normalization, the spectra and cospectra are each reduced to a family of curves which spread out according to z/L at low frequencies but converge to a single universal curve in the inertial subrange. The paper compares these results with data obtained by other investigators over both land and water.

Spectral constants for velocity and temperature are determined and the variability in the recent estimates of the constants is discussed. The high-frequency behaviour is consistent with local isotropy. In the inertial subrange, where the spectra fall as n−5/3, the cospectra fall faster: uω and ωθ as n−7/3, and uθ, on the average, as n−5/2. The 4/3 ratio between the transverse and longitudinal spectral levels is observed at wavelengths of the order of the height above ground under unstable conditions and at wavelengths of the order of L/10 under stable conditions. This lower isotropic limit is shown to be governed by the combined effects of shear and buoyancy on small-scale eddies.

Spectral Characteristics of Surface-Layer Turbulence

The sensitivity of sonic anemometer-derived stress estimates to the tilt of the anemometer is investigated. The largest stress errors are shown to occur for unstable stratification (z/L<0) and deep convective boundary layers. Three methods for determining the tilt angles relative to a mean streamline coordinate system and for computing the tilt-corrected stresses are then compared. The most commonly used method, involving a double rotation of the anemometers' axes, is shown to result in significant run-to-run stress errors due to the sampling uncertainty of the mean vertical velocity. An alternative method, requiring a triple rotation of the anemometer axes, is shown to result in even greater run-to-run stress errors due to the combined sampling errors of the mean vertical velocity and the cross-wind stress. For measurements over the sea where the cross-stream stress is important, the double rotation method is shown to overestimate the surface stress, due to the uncorrected lateral tilt component. A third method, using a planar fit technique, isshown to reduce the run-to-run stress errors due to sampling effects, and provides an unbiased estimate of the lateral stress.

Sonic anemometer tilt correction algorithms

Most implementations of genetic algorithms experience sampling bias and are unnecessarily inefficient. This paper reviews various sampling algorithms proposed in the literature and offers two new algorithms of reduced bias and increased efficiency. An empirical analysis of bias is then presented.

Reducing bias and inefficiency in the selection algorithm

Presents, in a single volume, an up-to-date summary of the current knowledge of the statistical characteristics of atmospheric turbulence and an introduction to the methods required to apply these statistics to practical engineering problems. Covers basic physics and statistics, statistical properties emphasizing their behavior close to the ground, and applications for engineers.

Atmospheric Turbulence: Models and Methods for Engineering Applications

It is proposed that the ratios of the standard deviations of the horizontal velocity components to the friction velocity in the surface layer under convective conditions depend only onz

 i
 /L wherez

 i
 is the height of the lowest inversion andL is the Monin-Obukhov length. This hypothesis is tested by using observations from several data sets over uniform surfaces and appears to fit the data well. Empirical curves are fitted to the observations which have the property that at largez

 i
 /-L, the standard deviations become proportional tow
*, the convective scaling velocity. Fluctuations of vertical velocity obtained from the same experiments scale withz/L, wherez is the height above the surface, in good agreement with Monin-Obukhov theory.

The characteristics of turbulent velocity components in the surface layer under convective conditions

Spectra of atmospheric turbulence recently measured at various heights and sites under a variety of stability conditions have been analysed and compared. The results are: 

(i)In regions over which the spectra obey −5/3 power laws, the ratio of the lateral to the longitudinal spectra shows fair agreement with the 4/3 ratio predicted by the Kolmogorov hypothesis for the inertial sub-range. The vertical-longitudinal ratio has a similar tendency.
(ii)Dissipation rates computed from the longitudinal spectra seem to be consistent with the hypothesis that dissipation is balanced by the total production of mechanical and convective turbulent energy, provided that the turbulence is in equilibrium. In transition from rough to smooth terrain, dissipation exceeds the other terms.
(iii)Vertical-velocity spectra obey Monin-Obukhov similarity theory up to a height of about 50 m. Their shapes are reasonably uniform, the major change with stability being a change of scale of the wave number axis, i.e., any characteristic nondimensional wave number is a function of z/L only. This function appears to be the same as the relation between the normalized dissipation and z/L. These results are consistent with previously measured Kolmogorov constants and with measured ratios of standard deviation of vertical velocity to friction velocity. Up to about a height of 50 m the wavelengths of the maxima of the logarithmic spectra increase linearly with height and more slowly thereafter, up to about 300 m. The spectra in stable air above 50 m suggest the existence of a buoyant sub-range.
(iv)Longitudinal spectra do not obey similarity theory in a number of ways. The wavelengths do not scale with height, and there may be differences between sites when the spectra are plotted in similarity coordinates.
(v)Spectra over the sea seem to have relatively more energy at low frequencies than those over land.

Recent spectra of atmospheric turbulence

Determination of stress from wind and temperature measurements

To describe the distribution of wind with height in hydrostatically-neutral air following a change in terrain roughness, a theory is constructed by assuming that only the air below an internal boundary is affected by the change and that air above the boundary is moving with the speed and Reynolds stress that it had upwind of the change of roughness. Unlike in the closely related theory of Elliott, the assumed velocity distribution in the lower layer is consistent, not with stress independent of height, but with continuous variation of stress from the surface to the boundary where the stress is that in the original layer. the condition of overall conservation of momentum then leads to relations between surface stress, boundary height and fetch. the theory is compared with observations at a variety of locations, and the agreement with measurements is satisfactory. In particular, a fairly sharp boundary separates the masses of air influenced by the two types of terrain; for micrometeorological distances, the slope of this interface is of order 1/10.

Hans A. Panofsky

Papers

Atmospheric Turbulence: Models and Methods for Engineering Applications

The characteristics of turbulent velocity components in the surface layer under convective conditions

Recent spectra of atmospheric turbulence

Determination of stress from wind and temperature measurements

Change of terrain roughness and the wind profile