Munk Model Research Articles

Light penetration in soil and particulate minerals

SummaryKnowledge of light penetration in soils is of particular interest for photolytic degradation of pesticides, for laser‐induced fluorescence spectroscopy and remote sensing, and for understanding better the germination of seeds. To date little information has been available in the literature on this topic. In this paper light penetration in soils is determined successfully using diffuse reflectance and transmittance spectroscopy and the relatively simple Kubelka–Munk model. Using the latter model of light propagation in turbid media, the optical properties of kaolinite, montmorillonite, barium sulphate, goethite and 19 different soils were determined in the wavelength range 275–700 nm. In particular, the light absorption coefficient, k, and light scattering coefficient, s, were determined. The depth at which the light intensity at the surface is reduced by 99% (light penetration depth) and the depth of the sample contributing to the measured reflected radiation (information depth) could also be calculated from k and s. For kaolinite, the light penetration depth ranged between 10 and 200 µm for wavelengths between 275 and 700 nm, respectively; the information depth was between 5 and 80 µm. For soils, the penetration depth was in the range 17–110 µm at 275 nm and 120–300 µm at 700 nm, and the information depth was in the range 8–60 µm at 275 nm and 60–175 µm at 700 nm. Hence, the information depth is about half the penetration depth for media with reflectance smaller than 0.7 (e.g. soils). For dry soils, an empirical relationship was established between the light absorption coefficient and the amount of particle‐size fractions. The effect of water content was also investigated: addition of water to kaolinite reduced its scattering coefficients, whereas the absorption coefficient was hardly affected. For the soils, addition of water had a more complex mode of action affecting both absorption and scattering coefficients. With the measured optical properties of dry minerals and soils it is possible to calculate light intensity profiles with depth and to quantify photochemical processes occurring in these media.

Wind-driven barotropic gyre II: Effects of eddies and low interior viscosity

Using boundary-enhanced viscosity to control the mean circulation, a simple model can be created and used for study of strong inertial effects in a western-intensie ed calculation. The simplicity allows for a greater number of strongly-inertial numerical experiments than computationally feasible in a general circulation model. This paper is an introduction to the behavior of this model, covering its general features. Some of the inertial phenomena, including the primary balances of the boundary current and basin interior, the temporal behavior, and the changes in the mean state across parameter space are presented. The analysis of these phenomena focuses on the effects of eddies and the type of eddies present. The low interior viscosity allows for more pronounced eddy effects. As this model is intended for use in future studies, many of the diagnostic tools found to be useful here are likely to be reused effectively. Fox-Kemper and Pedlosky (2004) demonstrate that the circulation strength of the single-gyre can be controlled with lower interior viscosity by a combination of eddy vorticity transport and boundary-enhanced viscosity. Once this control is achieved, it is possible to produce calculations which remain western intensie ed at a much higher Reynolds number in the basin interior than is possible with constant viscosity. Changing the viscosity in an inertial model is signie cantly more interesting than changing the viscosity in a linear Munk model. In the latter case, the widths of boundary currents change, but the character of the solution does not. In an inertial model, the results of changing the viscosity may be qualitative as well as quantitative. This paper explores some ofthe behavior ofthe strongly inertial, western-intensie ed Munk model, and presents the most surprising and stimulating results found during this investigation. It is demonstrated that the boundary layer vorticity balance, the basin interior vorticity balance, and the temporal character are all qualitatively affected by lowering the interior viscosity. A

Energy Transfer within the Small-Scale Oceanic Internal Wave Spectrum

Abstract Three-dimensional numerical experiments are conducted to examine energy transfer within the small-scale portion of the Garrett–Munk model spectrum of oceanic internal waves. The rate of energy transfer in the experiments is a significant fraction of observed total transfer rate in the interior main thermocline. This transfer may supplement the previous estimate by the eikonal theory. Because nonlinearity is strong in this spectral region, wavenumber-local interactions dominate the energy transfer rather than scale-separated ones. Transfer to higher horizontal wavenumbers is robust, whereas that to higher vertical wavenumbers seems to depend strongly on the spectral shape. Vortical motions seem to be enhancing energy transfer. All of these suggest that further investigation of this spectral region is important and necessary by means of three-dimensional, fully nonlinear analysis.

Slope Control in Western Boundary Currents

An analytic solution is presented for the steady-state depth-averaged western boundary current flowing over the continental slope by combining three highly idealized models: the Stommel model, the Munk model, and the arrested topographic wave model. The main vorticity balance over the slope is between planetary vorticity advection and the slope-induced bottom stress torque, which is proportional to ry (h21)x where r is the Rayleigh friction coefficient, h is the water depth, and y is the meridional velocity. This slope-induced torque provides the necessary source of vorticity for poleward flow over the slope, its simple interpretation being that vorticity is produced because the bottom stress has to act over the seaward-deepening water column. The character of the solution depends on the slope a as well as on the assumed bottom drag coefficient, and the length scale of the boundary current is ; . It is further shown that, if the depth-averaged velocity flows along isobaths, ˇ2r/(ba) then the stretching of water columns associated with cross-isobath geostrophic flow, which compensates bottom Ekman transport, is identical to the slope-induced torque by the geostrophic velocities.

Correction of sun glint contamination on the SeaWiFS ocean and atmosphere products

For remote sensing of the ocean and atmosphere optical properties, the measurement of radiances affected by sun glint has to be avoided and/or masked out. There are usually no meaningful retrievals in regions significantly contaminated by sun glint. The Sea-viewing Wide Field-of-view Sensor (SeaWiFS) is capable of operationally tilting the sensor 20 degrees away from nadir to minimize sun glint contamination. The sun glint mask is computed from the Cox and Munk model [J. Opt. Soc. Am. 44, 838-850 (1954)] and applied to the SeaWiFS data. However, sun glint is still a factor near the subsolar point. We present results that demonstrate the effect of sun glint contamination on retrievals of ocean bio-optical and atmospheric products. We show that, although sun glint contamination has a minor effect on retrieved ocean color products, the effect on retrieved atmospheric products, e.g., aerosol optical thickness, is significant. We describe a sun glint correction scheme implemented in the SeaWiFS data-processing system and compare the results with and without sun glint correction. With sun glint correction the derived ocean and atmospheric products are improved. Also, the sun glint masked area can be reduced and therefore can increase significantly the coverage area near the subsolar point.

Internal wave effects on ocean acoustic propagation in shallow water

Complementary models are developed to investigate the effect of internal waves on ocean acoustic transmissions in shallow water. An empirical model, based on temperature variations, and a modified Garrett–Munk spectral model of internal wave generated sound speed variations are in turn inserted into an acoustic transmission model. The Garrett–Munk model is itself an empirical model most often used in deep water; in shallow water a different parametric fit is needed to model the complexity found in the environment. The internal wave effects on the acoustic field and, in particular, the patterns of the field extrema, obtained from the numerical modelling are then compared with the results of 2–3 and 5–7 kHz broadband transmissions collected during a shallow water acoustics experiment off the coast of San Diego. The two modelling approaches are used to gain insight into the effects of internal waves on the motion of dislocations; dislocations are places in the sound field where the amplitude goes to zero and so the phase becomes undetermined. [Work supported by ONR.]

An approach to the solution of multidimensional problems of the theory of light scattering in turbid media

A new approach to developing simple analyticalmodels in multidimensional theory of light scattering in turbidmedia is proposed. The approach generalises the one-dimensional Kubelka — Munk model to the cases of two and threespatial dimensions, allowing one to solve some problemsexplicitly with the precision that is sufficient for practicalapplications.

Characterization of the internal waves and its effect on signal propagation in the Adventure Bank area

During the Advent-99 experiment, which took place in the Sicily Strait in May, 1999, multitone (in the band 200–1500 Hz) narrow-band signals were transmitted from a bottom-mounted tower and received on a 64 phone line array (VLA) covering the water column; the water depth was 80 m. Data at a source–receiver range of 10 km will be presented and modeled using an internal wave model. A CTD chain containing 48 CTDs on a vertical string was towed continuously during the acoustic experiments between the source and VLA, yielding a time-evolving sound speed profile between the source and receiver. From that the average buoyancy frequency and sound speed profile were determined as a function of range. Internal-wave-mode depth functions were calculated using the measured buoyancy profiles. Internal-waves-mode amplitudes were determined from the eigenvalues of the sound speed covariance matrix for the lowest 5 modes. Internal-wave-frequency spectrum has the typical 2-power dependence as in the Garrett–Munk model. The modeled results of mean transmission loss and amplitude fluctuations with and without internal waves will be presented and compared with data. [Work supported by the Office of Naval Research and SACLANTCEN.]

Atmospheric correction for the satellite visible data over heterogeneous surfaces

An atmospheric correction algorithm over the heterogeneous surface on the emergent radiation from the top of the atmosphere is proposed. The surface is simulated by checkerboard type terrain composed of land or ocean pixels. The look-up-table method is used for the atmospheric correction. To investigate the contribution of adjacent pixels, two additional parameters based on the diffuse transmission and reflection function of the atmosphere are introduced. These parameters are independent of the surface reflection properties. At present, the ocean surface is in accordance with the Cox and Munk model surface, whereas that of land is assumed to be Lambertian. The upwelling radiation emerging from the top of the atmosphere is expressed by a sum of radiative interactions between the surface and the atmosphere, and it is simulated using these parameters and combinations until convergence. The present method was developed with reference to currently existing numerical results based on the one-dimensional radiative transfer code. An example of numerical simulation is shown at a wavelength of 0.56 μm, which corresponds to the representative wavelength of the visible region of spectrum. The dust-like model is used for the aerosols, and the land surface albedo is 0.2 or 0.4. The isotropic ocean wind is 5m/s, and the refractive index is 1.333. The effect of the land surface on the radiance over the ocean is stronger just off the coastal zone and decreases exponentially with increasing distance from the land. It depends upon the solar zenith angle and atmospheric condition as well. The present new version enables us to quantitatively discuss radiative transfer over the heterogeneous surface including the coastal zone. A typical numerical simulation is performed over the Tsukuba lake area in Japan.

The Influence of Negative Viscosity on Wind-Driven, Barotropic Ocean Circulation in a Subtropical Basin

Abstract In this paper, a new barotropic wind-driven circulation model is proposed to explain the enhanced transport of the western boundary current and the establishment of the recirculation gyre. In this model the harmonic Laplacian viscosity including the second-order term (the negative viscosity term) in the Prandtl mixing length theory is regarded as a tentative subgrid-scale parameterization of the boundary layer near the wall. First, for the linear Munk model with weak negative viscosity its analytical solution can be obtained with the help of a perturbation expansion method. It can be shown that the negative viscosity can result in the intensification of the western boundary current. Second, the fully nonlinear model is solved numerically in some parameter space. It is found that for the finite Reynolds numbers the negative viscosity can strengthen both the western boundary current and the recirculation gyre in the northwest corner of the basin. The drastic increase of the mean and eddy kinetic en...

Measurement of modal spectrum of linear internal waves in shallow water and comparison with the Garrett–Munk model

The properties of linear internal waves are well studied in deep ocean and are surprisingly well modeled by the Garrett–Munk model. The properties of linear internal waves in shallow water are not well studied. Experimental measurements may be hindered by the contributions of nonlinear internal waves, the complications caused by range-dependent coastal oceanography, and possible interactions with the bottom boundary layers. There is no reason to expect the Garrett–Munk model to work in shallow water, as some of the assumptions made in the model may be violated. In this paper, the modal spectrum (the modal amplitudes) of the linear internal waves are deduced from approximately 2 h of CTD data collected during the SWARM 95 experiment during a period when nonlinear internal waves are absent or negligible. The method of empirical orthogonal decomposition is applied to the covariance matrix of the sound speed variations assuming that the internal waves modes are uncorrelated. One finds that the eigenvectors of this matrix agree well with the theoretical eigenfunctions of the first five modes of the internal waves. The mode amplitudes are deduced from the eigenvalues and compared with the Garrett–Munk model. The measured mode amplitudes agree with the measurements using a different method which projects the sound speed variations onto the theoretically calculated mode depth functions. [This work is supported by the Office of Naval Research.]

Frequency spectrum of linear internal waves in shallow water and a modified Garrett–Munk model

The properties of linear internal waves are well studied in deep ocean and are surprisingly well modeled by the Garrett–Munk model. In the Garrett–Munk model, the internal wave spectrum is the product of the modal spectrum and the wave-number spectrum. The properties of linear internal waves in shallow water are not well studied. In this paper the mode amplitude and frequency spectrum are deduced using the mode decomposition method based on the theoretical mode depth functions. The mode decomposition method is applied to the sound speed variation data as a function of time and depth. Advantages of using sound speed variation data versus the particle displacement data are discussed. The mode amplitude measurements agree with the results using the empirical orthogonal decomposition approach. A modified Garrett–Munk model is proposed to explain the deviation of the spectrum slope from the original Garrett–Munk model. [This work is supported by the Office of Naval Research.]

Attenuation and forward scattering of short-duration laser radiation in a strongly scattering medium

An experimental investigation was made of the dependence of the optical density of a layer of a strongly scattering medium on its thickness in the case of a model aqueous milk solution. The radiation sources were IR lasers generating pulses of various (from nanosecond to femtosecond) durations. The dependences of the attenuation coefficients of such pulses on the solution concentration were determined for low and high optical densities of the solution layer. Estimates were obtained of the absorption and scattering coefficients of the medium. A modification of the two-flux Kubelka — Munk model was used to derive an expression describing the dependence of the transmission of a solution layer on its parameters when radiation detectors with a finite angular aperture are used. A comparison of the calculations and experiments revealed a forward scattering anisotropy of short-duration laser radiation in an aqueous milk solution.

On the stability of wind-driven barotropic ocean circulations

Using numerical simulations, the circulation patterns of wind-driven barotropic oceans are re-explored. The classical Munk and Stommel models are revisited and extended by including non-linear phenomena. Emphasis is placed on the definition of stable flow régimes and an attempt is made to draw up a general scheme of these in the parameter space defined by δi (non-linearity), δl (lateral friction) and δb (bottom friction). Examples of configurations arising in the special case of partial slip boundary conditions are also discussed.

Statistics of Shear and Turbulent Dissipation Profiles in Random Internal Wave Fields

Abstract Because breaking internal waves produces most of the turbulence in the thermocline, the statistics of ϵ, the rate of turbulent dissipation, cannot be understood apart from the statistics of internal wave shear. The statistics of ϵ and shear are compared for two sets of profiles from the northeast Pacific. One set, PATCHEX, has internal wave shear close to the Garrett and Munk model, but the other set, PATCHEX north, has average 10-m shear squared, 〈S210〉, about four times larger than the model. The 10-m, shear components, Sx and Sy, were measured between 1 and 9 MPa and referenced to a common stratification by WKB scaling. The scaled components, Sˆx and Sˆy, are found to be independent and normally distributed with zero means, as assumed by Garrett and Munk. This readily leads to analytic forms for the probability densities of Sˆ210 and Sˆ410. The observed probability densities of Sˆ210 and Sˆ410 are close to the predicted forms, and both are strongly skewed. Moreover, σlnSˆ210 and σlnSˆ410 are c...

Varieties of Fully Resolved Spectra of Vertical Shear

The Multi-Scale profiler (MSP) resolves shear between vertical wavenumbers of 0.01 cpm and the viscous cutoff of small-scale turbulence. Observations from five sites reveal varied spectral shapes and amplitudes. Spectral amplitudes measured at low latitude do not increase toward the equator, contrary to Munk, and their shapes differ from the Garrett and Munk model by having weak maxima between 0.02 and 0.05 cpm. Moreover, at wavenumbers larger than 0.1 cpm these spectra roll off more steeply, k3−1, than do spectra at midlatitude. Of two average spectra from midlatitude, one is close to the Garrett and Munk model at low wavenumbers, and at 0.1 cpm it begins to roll off as k3−1. The second midlatitude spectrum has amplitudes well above Garrett and Munk at low wavenumbers, begins a k3−1 rolloff near 0.04 cpm, and has a well-developed turbulent range near 1 cpm. The decrease in the start of the rolloff is not linearly proportional to the increase in spectral amplitude at low wavenumber, unlike the spectra observed by Duda and Cox and models proposed by Munk and also Garrett. In spite of the diversity of shapes and amplitudes at low wavenumbers, all shear spectra have nearly the same amplitude at 0.14 cpm, which is in the rolloff range. The rolloff range cannot be a buoyancy subrange of three-dimensional turbulence because the largest overturns occur only at the high-wavenumber end of the range. Rather, the rolloff must be the signature of the high-wavenumber decay of the internal wave field. Near 0.5kE = (N3/ε)1/2 spectra change from the internal wave rolloff to the turbulent dissipation range, which is adequately represented by Nasmyth's “universal” spectrum. Midlatitude spectra with amplitudes close to the Garrett and Munk model have very weak turbulent spectra, but those with substantially larger low-wavenumber amplitudes have well-developed turbulent spectra with distinct inertial subranges. Owing to their steeper rolloffs, the low-latitude records also have weak dissipation spectra even though their spectra rise above Garrett and Munk at wavenumbers slightly less than the rolloff.

A statistical mechanical explantion of the Garrett and Munk model of oceanic internal waves: Allen, K.R. and R.I. Joseph, 1989. Johns Hopkins APL tech. Dig., 10(4):348–361

A statistical mechanical explantion of the Garrett and Munk model of oceanic internal waves: Allen, K.R. and R.I. Joseph, 1989. Johns Hopkins APL tech. Dig., 10(4):348–361

Scaling turbulent dissipation in the thermocline

By comparing observations from six diverse sites in the mid‐latitude thermocline, we find that, to within a factor of 2, , where 〈εIW〉 is the average dissipation rate attributable to internal waves; N0 = 0.0052 s−1 is a reference buoyancy frequency; S10 is the observed shear having vertical wavelengths greater than 10 m; and SGM is the corresponding shear in the Garrett and Munk spectrum of internal waves. The functional form agrees with estimates by McComas and Müller and by Henyey, Wright, and Flatté of the rate of energy transfer within the internal wave spectrum, provided the energy density of the internal waves is treated as a variable instead of one of the constant parameters. Following Garrett and Munk, we assume that , where EIW is the observed energy density and EGM is the energy density used by Garrett and Munk. The magnitude of εIW is twice that of Henyey et al. and one third that of McComas and Müller. Thus the observations agree with predictions sufficiently well to suggest that (1) a first‐order understanding of the link between internal waves and turbulence has been achieved, although Henyey et al. made some ad hoc assumptions and Garrett and Munk's model does not match important features in the internal wave spectrum reported by Pinkel, and (2) the simplest way to obtain average dissipation rates over large space and time scales is to measure . Even though the observations were taken at latitudes of 42°−11.5°, the variability in the Coriolis parameter ƒ was too limited for a conclusive test of the ƒ dependence also predicted for 〈εIW〉 by the wave‐wave interaction models. An exception to the scaling occurs east of Barbados in the thermohaline staircase that is apparently formed and maintained by salt fingers. Although ε in the staircase is very low compared with rates at mid‐latitude sites, it is 8 times larger than predicted for ε due only to internal waves.

The dependence of turbulent dissipation on stratification in a diffusively stable thermocline

To relate the viscous dissipation rate ϵ in the thermocline to internal waves, we operated the multiscale profiler (MSP) in the eastern North Pacific. The location is well removed from fronts and strong currents, and below 0.6 MPa the water column is diffusively stable. Hence internal wave breakdown is the only known source of turbulence. We found that shear levels were in close agreement with the Garrett and Munk model of internal waves and that dissipation rates were low, . This is the first report of > (previous observations indicated ) and agrees with predictions by McComas and Müller (1981a) and by Henyey et al. (1986), both based on calculations of energy transfer within the internal wave spectrum.

Intensity fluctuations. Part II: Comparison with the Cobb experiment

The intensity fluctuations measured at 4 and 8 kHz in the Cobb experiment have been available for nearly 10 years and in that time have not successfully been predicted theoretically. We show that multiple scatter effects must be considered, and that neither the Born nor the Rytov approximation to the scattering formulation is appropriate. A companion paper [J. Acoust. Soc. Am. 74, 1474–14832 (1983)] provides the theoretical background for this work by presenting a general form of the analytical solution to the fourth moment equation in two dimensions for a point source transmission. We use the parameters of the Garrett–Munk model of the internal wave field appropriate to the Cobb experiment oceanographic regime to obtain the correlation functions of the acoustic refractive index field, and then predict the intensity fluctuations. We discuss corrections to the predicted spectrum that are due to fine structure effects and tidal motions. We include a discussion of the scattering parameters Γ and X, which are the scattering and range scaling parameters of the moment equation formulation, and the parameters of the path integral formulation, Γ and Φ. We present a discussion of the measured correlation functions taken during a more recent experiment at Cobb seamount and indicate the directions we need to take to bring the scattering predictions based on internal waves and finestructure into closer agreement with experiment.