Gradient Richardson Number Ri Research Articles

On the role of shear mixing during transient coastal upwelling

A two-dimensional two-layer model for wind-driven transient coastal upwelling is formulated. The momentum equations include the turbulent dynamics and the time-dependent and nonlinear terms in both the cross- and along-shore directions. The continuity and heat equations allow mass and heat turbulent transfer between both layers. The integral form of the momentum, continuity and heat equations are closed using a two-regime parameterization for the entrainment velocity. In the first regime, corresponding to the early stages of upwelling, the interface quickly raises due to flux divergence near the coast. The entrainment velocity is small (0.1-1 m day −1, largely produced through K raus and T urner's [(1967) Tellus, 19, 98–106] slow erosion of the thermocline, and it is estimated using N iiler and K raus' [(1977) In: Modelling and prediction of the upper layers of the ocean, Pergamon Press, Oxford, pp. 143–172] parameterization. When the bulk Richardson number ( Ri) becomes close to its critical value then we switch to the second regime, during which we calculate the entrainment velocity from the continuity equation under the condition that Ri remains near-critical, i.e. the equivalent of P ollard et al. [(1973) Geophysical Fluid Dynamics, 4, 381–404] stability criterion for the upper ocean. The entrainment velocity quickly becomes large (several m h −1), the interface deepens and stratification is eroded. The existence of this regime is supported by observations of persistent near-critical gradient Richardson numbers ( Ri g ) during coastal upwelling [J ohnson (1981) In: Coastal upwelling, American Geophysical Union, Washington, DC., pp. 79–86; J ohnson et al. (1976) Journal of Physical Oceanography, 6, 556–574; K undu and B eardsley, (1991) Journal of Geophysical Research, 96, 4855–4868]. Our model is applied to several initial temperature differences between the surface and bottom layers, with the upper layer depth and forcing parameters realistically chosen. The dynamically important mixing regime corresponds to the second regime, with effective shear-induced mixing being produced through a strong baroclinic coastal jet. A realistic front, formed between the well-mixed water near the coast and lighter offshore surface water, propagates away from the coast. The offshore waters are characterized by the presence of inertial oscillations, overlying the Ekman flow. The inertial oscillations are too weak to produce any significant mixing, but a comparison with de S zoeke and R ichman's [(1984) Journal of Physical Oceanography, 14, 364–377] semigeostrophic model (modified to include the shear-mixing regime) shows that they are important enough to exert some control on the horizontal volume flux divergence near the coast. A relatively fast internal Poincare wave, propagating from the coast, has the effect of slowly dampening the inertial oscillations. The results are in good qualitative agreement with early observations by J ohnson et al. (1976).

Observations of the second vertical mode of the internal seiche in an alpine lake

The analysis of wind and temperature data from Alpnacher See (Switzerland) shows the second vertical mode of the internal (baroclinic) seiche to dominate over the first vertical mode. Resonance with diurnal wind is responsible for the high amplitudes of the second mode. Two different models are used to calculate the periods and velocity distribution of the various modes of the internal seiche. First, as a natural extension of Mortimer’s two‐layer model, the eigenvalues of a three‐layer model for a rectangular basin are calculated analytically. The period of the second vertical mode is found to depend strongly on the thickness of the metalimnion. Second, in order to better represent the vertical structure of the water column, a continuous stratification model is adopted and solved analytically for realistic density profiles. The resulting vertical variation of the horizontal velocity is used to calculate local gradient Richardson numbers (Ri) and relate them to the vertical eddy diffusivity estimated with the temperature microstructure method.

A simple description of the deepening and structure of a stably stratified flow driven by a surface stress

The deepening and structure of a stably stratified flow driven by a surface stress are examined by means of a simple conceptual model, in which local mixing does not occur if the gradient Richardson number Ri is greater than the critical value Ric ≃ ¼ indicating onset of shear instability, and local mixing occurs instantaneously, through a simple exchange of fluid volumes, if Ri is less than Ric. The model leads to a mathematical formulation in which turbulent mixing is a gradient transport process just strong enough to maintain Ri at the critical value Ric throughout the boundary layer. Although unrealistic just beneath the stressed surface, self‐similar solutions based on this formulation plausibly describe the outer part of the boundary layer and the rate at which the layer deepens, provided that appropriately defined bulk Richardson and Reynolds numbers are sufficiently large. A comparison of model computations with existing laboratory measurements and turbulence closure simulations supports the conceptual model as an approximate representation of the processes controlling the deepening of the boundary layer and the qualitative structure in the outer part of the flow. Estimates are obtained of the region within the boundary layer and in parameter space where the formulation is realistic.

On the characteristics of eddies in the stable atmospheric boundary layer

A discussion of the cross-spectral properties of eddies in the lowest 40 m of the nocturnal boundary layer is presented. The study involves the analysis of meteorological data collected by the British Antarctic Survey at Halley Station, Antarctica, during the austral winter of 1986. Cross-spectral analysis is used to determine whether the nature of the observed eddies is primarily turbulent or whether their structure is characteristic of coherent internal gravity waves. It is found that the cross-spectral phases indicate the presence of turbulent eddies only when the local gradient Richardson number (Ri) is less than the critical value of 1/4. Trapped modes were only observed when an off-shore wind prevailed, indicating that topographic effects are responsible for their generation. The relative phases of velocity and temperature were often observed to change with height. This can be explained by examining the underlying meteorological conditions. On several occasions, regions of counter-gradient fluxes were detected. A physical explanation of this phenomenon is proposed.

On the Analytical Solutions of Flux-Profile Relationships for the Atmospheric Surface Layer

Analytical solutions Of surface layer similarity equations are presented for the nondimensional profile functional forms suggested by Businger et al. For stable atmospheric conditions, exact analytical solutions of the Monin-Obukhoy stability parameters ζ=z/L, in terms of gradient Richardson number (Ri) and bulk Richardson number (Rib), respectively, are obtained. For unstable atmospheric conditions, an exact solution of the same in terms of Ri is found. An approximate relationship between ζ and Rib, for an unstable surface layer, based on the solution of ζ in terms of Ri, is suggested. The proposed analytical solutions can replace costly numerical iterative methods as well as some of the ad hoc parameterizations used for estimating surface fluxes and related turbulence characteristics.

Direct numerical simulation of stratified homogeneous turbulent shear flows

The exact time-dependent three-dimensional Navier-Stokes and temperature equations are integrated numerically to simulate stably stratified homogeneous turbulent shear flows at moderate Reynolds numbers whose horizontal mean velocity and mean temperature have uniform vertical gradients. The method uses shear-periodic boundary conditions and a combination of finite-difference and pseudospectral approximations. The gradient Richardson number Ri is varied between 0 and 1. The simulations start from isotropic Gaussian fields for velocity and temperature both having the same variances.The simulations represent approximately the conditions of the experiment by Komori et al. (1983) who studied stably stratified flows in a water channel (molecular Prandtl number Pr = 5). In these flows internal gravity waves build up, superposed by hot cells leading to a persistent counter-gradient heat-flux (CGHF) in the vertical direction, i.e. heat is transported from lower-temperature to higher-temperature regions. Further, simulations with Pr = 0.7 for air have been carried out in order to investigate the influence of the molecular Prandtl number. In these cases, no persistent CGHF occurred. This confirms our general conclusion that the counter-gradient heat flux develops for strongly stable flows (Ri ≈ 0.5–1.0) at sufficiently large Prandtl numbers (Pr = 5). The flux is carried by hot ascending, as well as cold descending turbulent cells which form at places where the highest positive and negative temperature fluctuations initially existed. Buoyancy forces suppress vertical motions so that the cells degenerate to two-dimensional fossil turbulence. The counter-gradient heat flux acts to enforce a quasi-static equilibrium between potential and kinetic energy.Previously derived turbulence closure models for the pressure-strain and pressure-temperature gradients in the equations for the Reynolds stress and turbulent heat flux are tested for moderate-Reynolds-number flows with strongly stable stratification (Ri = 1). These models overestimate the turbulent interactions and underestimate the buoyancy contributions. The dissipative timescale ratio for stably stratified turbulence is a strong function of the Richardson number and is inversely proportional to the molecular Prandtl number of the fluid.

Mixing in the equatorial surface layer and thermocline

Twelve days of microstructure measurements at the equator (140°W) in November 1984 showed a surprisingly strong effect of both the daily cycle of solar heating and wind on mixing in the upper ocean. Because of limited variations in atmospheric forcing and currents during the experiment, processes in the daily mixing cycle were similar from day to day. Only the intensity of mixing varied. The lower boundary of the diurnal surface layer separated two distinct mixing regimes, the diurnal surface layer and the thermocline. Within the diurnal surface layer (which extended to 10‐ to 35‐m depth), turbulent kinetic energy dissipation rates ε varied relatively little. Although variations in surface layer depth coincided with the daily change in direction of air‐sea surface buoyancy production of turbulent kinetic energy (or simply, the surface buoyancy flux), ε was significantly greater relative to the buoyancy flux than was expected for a simple convective layer. In the thermocline below the diurnal surface layer, ε was highly intermittent; the day‐night cycle was stronger, and variability was enhanced by turbulent “bursts” of 2–3 hours duration, which may be related to internal wave breaking events. The turbulent heat flux crossing 20‐m depth was almost equal to the surface heat flux less the irradiance penetrating below 20 m. Seventy percent of the surface heat flux was transported vertically to the water below 30 m by turbulent mixing. Only a negligible amount penetrated to the core of the Equatorial Undercurrent. The gradient Richardson number Ri distinguishes between statistically different mixing environments. However, ε cannot be predicted from the value of Ri, since the intensity of mixing depends on the intensity of forcing in a way not specified by the value of Ri alone.

Growth and decay of turbulence in a stably stratified shear flow

The behaviour of an evolving, stably stratified turbulent shear flow was investigated in a ten-layer, closed-loop, salt-stratified water channel. Simultaneous single-point measurements of the mean and fluctuating density and longitudinal and vertical velocities were made over a wide range of downstream positions. For strong stability, i.e. a mean gradient Richardson number Ri greater than a critical value of Ricr ≈ 0.25, there is no observed growth of turbulence and the buoyancy effects are similar to those in the unsheared experiments of Stillinger, Helland & Van Atta (1983) and Itsweire, Helland & Van Atta (1986). For values of Richardson number less than Ricr the turbulence grows at a rate depending on Ri and for large evolution times the ratio between the Ozmidov and turbulent lengthscale approaches a constant value which is also a function of Richardson number.Normalized velocity and density power spectra for the present experiments conform to normalized spectra from previous moderate- to high-Reynolds-number studies. With increasing or decreasing stability, the stratified shear spectra exhibit greater portions of the universal non-stratified spectrum curve. The shapes of the shear-stress and buoyancy-flux cospectra confirm that they act as sources and sinks for the velocity and density fluctuations.

The vertical structure of density and turbidity currents: Theory and observations

A simple numerical model is developed that successfully simulates observations of small‐scale, laboratory, density currents flowing down inclines of constant slope. The model results suggest that laboratory determinations of the bulk Richardson number have been biased by molecular processes but that determinations of the entrainment coefficient are probably applicable to large‐scale currents, and even to turbidity currents in which the gravitational driving force is provided by suspended sediment. The entrainment coefficient is a function of the gradient Richardson number Ri above the velocity maximum in both density and turbidity currents; Ri is itself a strong function of the bottom slope but not of the sediment settling velocity or the roughness of the bottom boundary. Therefore for a given bottom slope the entrainment coefficient is uniquely defined. The bulk Richardson number Ri0, on the other hand, is additionally a function of the drag coefficient, which depends on both the Reynolds number and the bottom roughness. However, Ri0 is not a strong function of the sediment settling velocity as long as the current is energetic enough to maintain sediment in suspension.

Mixing efficiency in stably stratified growing turbulence

Nearly steady state mixing efficiencies, as characterized by the flux Richardson number Rf, have been measured in stably stratified, turbulent, uniform gradient shear flows. The turbulent dissipation rate ϵ, the vertical mass and momentum fluxes and , and the mean and rms density and velocity fluctuations were obtained from simultaneous single point measurements of velocity and density. Rf and the gradient Richardson number Ri were calculated at a number of positions in the spatially evolving shear flow. Within the investigated range of stabilities 0.02<Ri<0.08, Rf was observed to increase with Ri. It is found that the length scale arguments successfully applied by Rohr et al. (1984) to the case of decaying, stably stratified, unsheared grid turbulence can also account for the overall behavior of the mixing efficiency (Rf) as a function of stability (Ri) in the present case of growing, stably stratified, uniform mean shear turbulence.

Wind mixing of a stratified shear flow

The response of a shear flow to an imposed wind stress is studied both theoretically and by means of a numerical turbulence model. It is shown that for small initial gradient Richardson numbers ( Ri 0 ≲ 4/3 ) a tail wind causes the slab velocity of the upper mixed layer to decrease. The theory is based on the assumption that during the wind-induced entrainment process the overall Richardson number will adjust to a quasi-constant value ( Ri u ≈ 2/3 ). The turbulence model is the so-called k-ɛ model. It is calibrated to five conditions by tuning only one constant. The details of the deepening process and the density and velocity distributions of the upper mixed layer during this anomalous behavior are thus made clear. The results imply that the common practice of estimating the total current velocity by vector addition of the original velocity and the wind-induced velocity (calculated from models based on an ocean at rest) may lead to an overestimation of the current speed.

Formation of thermoclines in zero‐mean‐shear turbulence

Heating q at the surface of a turbulent and initially unstratified ocean may form a thermocline if strong enough. It is proposed here that the heating is strong enough if turbulence occurs infrequently enough past some depth not to transport heat downward. Turbulence is taken not to occur if the gradient Richardson number Ri = N2/(uz2 + Vz2) rises above a critical value of 0.25. The probability that this occurs becomes very small when and where the population Richardson number is greater than about 1.33. This traps the heat above that layer, increasing and forming a thermocline. Let stirring action K be generated at a depth D below heating, and scale the heating as B = gαq/ρcp. Then the population Richardson number criterion translates: if H = BD4/K3 < Hc ≈ 1.6, a thermocline will form above the stirring (“tidal friction”). If H < Hc, no thermocline will form above the stirring, but after a time 0.9(K/B)1/2, the thermocline will form at a depth 0.6D+2.2(K3/B)1/4 below the stirring (“wave breaking”). These theoretical results match experimental results of Hopfinger and Linden (1982).

Turbulence structure in unstably-stratified open-channel flow

The buoyancy effects on the turbulence structure in unstably-stratified shear flow generated by cooling the top of the layer in an open channel were investigated. Measurements were made of the characteristics of the turbulent fluctuations; turbulence intensities, correlation coefficients, joint probability density functions, and coherence-phase relationships. The results show that the buoyancy-driven motions cause substantial changes in the turbulence structure. Of the buoyancy-driven motions, the intermittent downward motion contributes mainly to the turbulent transport of heat in the vertical direction. A parameter which characterizes the stability dependence of the turbulence structure is the local gradient Richardson number Ri, and the turbulence quantities are well correlated with Ri. A theoretical spectral equation model is applied to the stratified flow and well compared with the experimental results.

Patterns in the Occurrences of Richardson Numbers Less Than Unity in the Lower Atmosphere

It is assumed that the presence of turbulence in the atmosphere is strongly correlated with low values of the gradient Richardson number (Ri ≤ 1.0). The critical Richardson number, Ric = 1.0, is an indicator of the onset of turbulence. Values of the gradient Richardson number are calculated using rawinsonde measurements of wind and temperature obtained from 144 rawinsonde stations from 1971 to 1975. Computer algorithms are developed for a statistical analysis of the critical Richardson number based on years, latitude, longitude, altitude and season. Twice-daily sets of rawinsonde height measurements of wind and temperature are fit using a Hermite interpolation algorithm. Richardson numbers calculated from these profiles are analyzed to determine the number and percent of occurrences of Ri ≤ Ric on a seasonal basis. Summary (seasonal) statistics on this critical Richardson number, or turbulence indicator, as well as all variables used, are computed for 1 km altitude bins from 2 to 25 km, a range determined to provide good statistical accuracy and precision. Particular features in the height profiles of percent occurrences of Ri ≤ Ric appear with remarkable consistency and are dependent on season and latitude. A characteristic peak layer in the percent occurrence of Ri ≤ 1.0 near 10 km is found in data obtained from sites located from low to midlatitudes. The range in occurrence for this layer is from 40% in winter to 5% in summer. Multiple regression analyses are applied using data from 21 of the stations covering a region from latitude 25 to 48°N and longitude 69 to 123°W. The data are sectioned for separate analysis into four altitude regions: 2–7, 8–13, 14–19 and 20–25 km. The analyses demonstrate that the patterns of turbulence based on a critical Richardson number of 1 have a substantial component which is stable and reproducible from year to year. The regressions relate percent occurrences of Ri ≤ Ric to location (latitude, longitude), altitude and season; the coefficients of multiple correlation, for altitudes below 20 km, range from 0.62 to 0.78 (i.e., up to 60% variation in percent occurrences explained). Yearly variations increase the square of the multiple correlation coefficients only by a maximum of 0.003 (i.e., at most 0.3% variation explained).

Calculation of vertical profiles of lake‐averaged temperature and diffusivity in Lakes Ontario and Washington

By using statistical analysis of temperature observations and model calculations, the common parameterization of vertical eddy diffusivity (k) in terms of the gradient Richardson number (Ri) is simplified. The simpler formulation accurately reproduces seasonal variations in temperature profiles from Lakes Ontario and Washington. Energy arguments support use of the simplified parameterization and suggest use of a modified Ri in calculating k. The new expression for k resulted in more realistic estimates of thermocline k. The modeling of free convective mixing in Lake Ontario by heat conservation resulted in excessive temperatures in the hypolimnion during fall overturn.