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Robert Hallberg - One of the best experts on this subject based on the ideXlab platform.
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Time Integration of Diapycnal Diffusion and Richardson Number–Dependent Mixing in Isopycnal Coordinate Ocean Models
Monthly Weather Review, 2000Co-Authors: Robert HallbergAbstract:Abstract In isopycnal coordinate ocean models, diapycnal diffusion must be expressed as a nonlinear difference equation. This nonlinear equation is not amenable to traditional implicit methods of solution, but explicit methods typically have a time step limit of order Δt ⩽ h2/κ (where Δt is the time step, h is the isopycnal layer thickness, and κ is the diapycnal diffusivity), which cannot generally be satisfied since the layers could be arbitrarily thin. It is especially important that the diffusion time integration scheme have no such limit if the diapycnal diffusivity is determined by the local Richardson Number. An iterative, implicit time integration scheme of diapycnal diffusion in isopycnal layers is suggested. This scheme is demonstrated to have qualitatively correct behavior in the limit of arbitrarily thin initial layer thickness, is highly accurate in the limit of well-resolved layers, and is not significantly more expensive than existing schemes. This approach is also shown to be compatible wi...
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time integration of diapycnal diffusion and Richardson Number dependent mixing in isopycnal coordinate ocean models
Monthly Weather Review, 2000Co-Authors: Robert HallbergAbstract:Abstract In isopycnal coordinate ocean models, diapycnal diffusion must be expressed as a nonlinear difference equation. This nonlinear equation is not amenable to traditional implicit methods of solution, but explicit methods typically have a time step limit of order Δt ⩽ h2/κ (where Δt is the time step, h is the isopycnal layer thickness, and κ is the diapycnal diffusivity), which cannot generally be satisfied since the layers could be arbitrarily thin. It is especially important that the diffusion time integration scheme have no such limit if the diapycnal diffusivity is determined by the local Richardson Number. An iterative, implicit time integration scheme of diapycnal diffusion in isopycnal layers is suggested. This scheme is demonstrated to have qualitatively correct behavior in the limit of arbitrarily thin initial layer thickness, is highly accurate in the limit of well-resolved layers, and is not significantly more expensive than existing schemes. This approach is also shown to be compatible wi...
Alexey A Grachev - One of the best experts on this subject based on the ideXlab platform.
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The Critical Richardson Number and Limits of Applicability of Local Similarity Theory in the Stable Boundary Layer
Boundary-Layer Meteorology, 2013Co-Authors: Alexey A Grachev, Peter S. Guest, Christopher W. Fairall, Edgar L. Andreas, P. Ola G. PerssonAbstract:Measurements of atmospheric turbulence made over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA) are used to determine the limits of applicability of Monin–Obukhov similarity theory (in the local scaling formulation) in the stable atmospheric boundary layer. Based on the spectral analysis of wind velocity and air temperature fluctuations, it is shown that, when both the gradient Richardson Number, Ri , and the flux Richardson Number, Rf , exceed a ‘critical value’ of about 0.20–0.25, the inertial subrange associated with the Richardson–Kolmogorov cascade dies out and vertical turbulent fluxes become small. Some small-scale turbulence survives even in this supercritical regime, but this is non-Kolmogorov turbulence, and it decays rapidly with further increasing stability. Similarity theory is based on the turbulent fluxes in the high-frequency part of the spectra that are associated with energy-containing/flux-carrying eddies. Spectral densities in this high-frequency band diminish as the Richardson–Kolmogorov energy cascade weakens; therefore, the applicability of local Monin–Obukhov similarity theory in stable conditions is limited by the inequalities Ri
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the critical Richardson Number and limits of applicability of local similarity theory in the stable boundary layer
Boundary-Layer Meteorology, 2013Co-Authors: Alexey A Grachev, Peter S. Guest, Christopher W. Fairall, Edgar L. Andreas, Ola P G PerssonAbstract:Measurements of atmospheric turbulence made over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA) are used to determine the limits of applicability of Monin–Obukhov similarity theory (in the local scaling formulation) in the stable atmospheric boundary layer. Based on the spectral analysis of wind velocity and air temperature fluctuations, it is shown that, when both the gradient Richardson Number, Ri, and the flux Richardson Number, Rf, exceed a ‘critical value’ of about 0.20–0.25, the inertial subrange associated with the Richardson–Kolmogorov cascade dies out and vertical turbulent fluxes become small. Some small-scale turbulence survives even in this supercritical regime, but this is non-Kolmogorov turbulence, and it decays rapidly with further increasing stability. Similarity theory is based on the turbulent fluxes in the high-frequency part of the spectra that are associated with energy-containing/flux-carrying eddies. Spectral densities in this high-frequency band diminish as the Richardson–Kolmogorov energy cascade weakens; therefore, the applicability of local Monin–Obukhov similarity theory in stable conditions is limited by the inequalities Ri < Ri cr and Rf < Rf cr. However, it is found that Rf cr = 0.20–0.25 is a primary threshold for applicability. Applying this prerequisite shows that the data follow classical Monin–Obukhov local z-less predictions after the irrelevant cases (turbulence without the Richardson–Kolmogorov cascade) have been filtered out.
Jeffrey R. Koseff - One of the best experts on this subject based on the ideXlab platform.
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Direct measurements of flux Richardson Number in the nearshore coastal ocean
2016Co-Authors: Jeffrey R. Koseff, Ryan K. Walter, Jamie Dunckley, Michael E. Squibb, Brock Woodson, Geno Pawlak, Stephen G. MonismithAbstract:Direct measurements of flux Richardson Number in the nearshore coastal ocean Jeffrey R. Koseff 1 , Presenting Author, Ryan K. Walter 1,2 , Jamie F. Dunckley 1,5 , Michael E. Squibb 1 , C. Brock Woodson 1,3 , Geno Pawlak 4 , and Stephen G. Monismith 1 1 Dept. of Civil and Environmental Eng., Stanford University, Stanford, CA 94305 (Presenting Author Email Address: koseff@stanford.edu) 2 Physics Dept., California Polytechnic State University, San Luis Obispo, CA, 93407 3 COBIA Lab, College of Engineering, University of Georgia, Athens, GA, USA, 30602 4 Dept. of Mech. and Aerospace Eng., Univ. of California San Diego, La Jolla, CA, 92093 5 Electric Power Research Institute, Palo Alto, CA, 94304. ABSTRACT We conducted a set of three field experiments using an underwater turbulence tower to address the following questions: (1) “What are the flux Richardson Numbers (R f ) and vertical diffusivities (K ) in a highly turbulent region with varying stratification?” (2) “Is it valid to extend the Shih et al. [2005] flux Richardson Number (R f ) parameterization to field situations at higher turbulence activity Numbers G = 2 ?”. The experiments were conducted at three separate field sites in ! ν N Monterey Bay, CA; Eilat, Israel; and Mamala Bay, Hawaii using the same experimental platform, instrumentation, and analytical methods. Direct measurements of turbulent buoyancy fluxes and mixing efficiencies, with 10 2 < G < 10 7 , confirm the relationship for the flux Richardson Number R f suggested by Shih et al. [2005]. Additionally, the mixing efficiency Γ is likely to be up to an order of magnitude less than the commonly assumed value of 0.2 over a wide range of turbulence states. This result holds over a range of flow conditions (including presence of internal waves and bores) and environmental conditions (weak and strong stratification) across the three sites. INTRODUCTION Quantifying the vertical turbulent mixing of scalars such as heat, dissolved oxygen, and dissolved inorganic carbon in the ocean, as well as in estuaries and lakes, remains an ongoing challenge [Ivey et al., 2008]. Vertical diffusivity can be directly calculated from measurements of the buoyancy flux if appropriate instrumentation is available. However, due to the difficulty of measuring buoyancy fluxes in the field, the diffusivity is often calculated from parameterizations that involve more easily measured variables (such as the temperature variance), estimates of the dissipation (using for example microstructure profilers, Thorpe-scale density overturns from moored profilers, fine-scale parameterizations, etc.), and an assumed mixing efficiency, Γ [cf. Osborn, 1980; Waterhouse et al., 2014]. Parameterizations for the vertical diffusivity and the mixing efficiency themselves vary widely depending on the level of turbulence and stratification [Dunckley et al., 2012]. Using direct numerical simulations (DNS) of stratified turbulence, Shih et al. [2005] found that Γ expressed in terms of the flux Richardson Number, R f (see Eq. 5), is a VIII th Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016
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On the flux Richardson Number in stably stratified turbulence
Journal of Fluid Mechanics, 2016Co-Authors: Subhas K. Venayagamoorthy, Jeffrey R. KoseffAbstract:The flux Richardson Number (often referred to as the mixing efficiency) is a widely used parameter in stably stratified turbulence which is intended to provide a measure of the amount of turbulent kinetic energy that is irreversibly converted to background potential energy (which is by definition the minimum potential energy that a stratified fluid can attain that is not available for conversion back to kinetic energy) due to turbulent mixing. The flux Richardson Number is traditionally defined as the ratio of the buoyancy flux to the production rate of turbulent kinetic energy . An alternative generalized definition for was proposed by Ivey & Imberger (J. Phys. Oceanogr., vol. 21, 1991, pp. 650–658), where the non-local transport terms as well as unsteady contributions were included as additional sources to the production rate of . While this definition precludes the need to assume that turbulence is statistically stationary, it does not properly account for countergradient fluxes that are typically present in more strongly stratified flows. Hence, a third definition that more rigorously accounts for only the irreversible conversions of energy has been defined, where only the irreversible fluxes of buoyancy and production (i.e. the dissipation rates of and turbulent potential energy ( )) are used. For stationary homogeneous shear flows, all of the three definitions produce identical results. However, because stationary and/or homogeneous flows are typically not found in realistic geophysical situations, clarification of the differences/similarities between these various definitions of is imperative. This is especially true given the critical role plays in inferring turbulent momentum and heat fluxes using indirect methods, as is commonly done in oceanography, and for turbulence closure parameterizations. To this end, a careful analysis of two existing direct numerical simulation (DNS) datasets of stably stratified homogeneous shear and channel flows was undertaken in the present study to compare and contrast these various definitions. We find that all three definitions are approximately equivalent when the gradient Richardson Number . Here, , where is the buoyancy frequency and is the mean shear rate, provides a measure of the stability of the flow. However, when , significant differences are noticeable between the various definitions. In addition, the irreversible formulation of based on the dissipation rates of and is the only definition that is free from oscillations at higher gradient Richardson Numbers. Both the traditional definition and the generalized definition of exhibit significant oscillations due to the persistence of linear internal wave motions and countergradient fluxes that result in reversible exchanges between and . Finally, we present a simple parameterization for the irreversible flux Richardson Number based on that produces excellent agreement with the DNS results for .
P. Ola G. Persson - One of the best experts on this subject based on the ideXlab platform.
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The Critical Richardson Number and Limits of Applicability of Local Similarity Theory in the Stable Boundary Layer
Boundary-Layer Meteorology, 2013Co-Authors: Alexey A Grachev, Peter S. Guest, Christopher W. Fairall, Edgar L. Andreas, P. Ola G. PerssonAbstract:Measurements of atmospheric turbulence made over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA) are used to determine the limits of applicability of Monin–Obukhov similarity theory (in the local scaling formulation) in the stable atmospheric boundary layer. Based on the spectral analysis of wind velocity and air temperature fluctuations, it is shown that, when both the gradient Richardson Number, Ri , and the flux Richardson Number, Rf , exceed a ‘critical value’ of about 0.20–0.25, the inertial subrange associated with the Richardson–Kolmogorov cascade dies out and vertical turbulent fluxes become small. Some small-scale turbulence survives even in this supercritical regime, but this is non-Kolmogorov turbulence, and it decays rapidly with further increasing stability. Similarity theory is based on the turbulent fluxes in the high-frequency part of the spectra that are associated with energy-containing/flux-carrying eddies. Spectral densities in this high-frequency band diminish as the Richardson–Kolmogorov energy cascade weakens; therefore, the applicability of local Monin–Obukhov similarity theory in stable conditions is limited by the inequalities Ri
Ola P G Persson - One of the best experts on this subject based on the ideXlab platform.
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the critical Richardson Number and limits of applicability of local similarity theory in the stable boundary layer
Boundary-Layer Meteorology, 2013Co-Authors: Alexey A Grachev, Peter S. Guest, Christopher W. Fairall, Edgar L. Andreas, Ola P G PerssonAbstract:Measurements of atmospheric turbulence made over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA) are used to determine the limits of applicability of Monin–Obukhov similarity theory (in the local scaling formulation) in the stable atmospheric boundary layer. Based on the spectral analysis of wind velocity and air temperature fluctuations, it is shown that, when both the gradient Richardson Number, Ri, and the flux Richardson Number, Rf, exceed a ‘critical value’ of about 0.20–0.25, the inertial subrange associated with the Richardson–Kolmogorov cascade dies out and vertical turbulent fluxes become small. Some small-scale turbulence survives even in this supercritical regime, but this is non-Kolmogorov turbulence, and it decays rapidly with further increasing stability. Similarity theory is based on the turbulent fluxes in the high-frequency part of the spectra that are associated with energy-containing/flux-carrying eddies. Spectral densities in this high-frequency band diminish as the Richardson–Kolmogorov energy cascade weakens; therefore, the applicability of local Monin–Obukhov similarity theory in stable conditions is limited by the inequalities Ri < Ri cr and Rf < Rf cr. However, it is found that Rf cr = 0.20–0.25 is a primary threshold for applicability. Applying this prerequisite shows that the data follow classical Monin–Obukhov local z-less predictions after the irrelevant cases (turbulence without the Richardson–Kolmogorov cascade) have been filtered out.
