Critical Richardson Number Research Articles

Deep surface mixed layers on the continental shelf

Simultaneous action of internal tides and wind-driven shear may be responsible for very deep (≥100m) surface mixed layers on the continental shelf. The conventional two-layer model, which considers deepening due to interfacial shear, ignores the strong coherent shear due to internal tides on the shelf. This shear is important on deeper pycnoclines which are less susceptible to wind-driven shear. In addition, the interfacial dynamics should be scaled by H 1 H 2/H (where H 1 and H 2 are the upper and lower layer depths and H = H 1 + H 2) in order to account for the proximity of the pycnocline to the lower boundary. After consideration of shear-produced mixing due to wind-driven motions alone and internal tides alone, a linear analytical model is used to consider the effect of maximum interference between these two forcing mechanisms in the coastal ocean. Various Richardson number criteria are used to control onset and persistence of internal mixing events. An event recorded by a thermistor string off the southwestern coast of Africa provided motivation for and confirmation of this model.

Application of a snow cover energy and mass balance model in a balsam fir forest

The objective of this study was to adjust the parameters of the point energy and mass balance model of a snow cover developed by E. Anderson in 1976 when applied to a balsam fir forest. This physically based model was used to simulate the snow cover energy and mass balance during spring at Lac Laflamme (47°N, 71°W). The calibration of the model was done with the physical properties of snow and hourly outflow observed at a snow lysimeter in 1985 and 1986; the validation was performed with the 1987 data. For the three seasons simulated, the model yielded accurate predictions, particularly of hourly and daily outflows. As expected, the forest canopy limits latent and sensible heat transfers to the snow cover because of its effect on wind speed. The prediction of outflow was almost insensitive to variations to the roughness parameter and to the critical Richardson number but was moderately sensitive to most parameters related to liquid water retention and transmission. Lack of fit between predicted and observed outflows, densities and temperatures at various levels in the snowpack occurred when ice layers or ice lenses were suspected to be present in the snow.

Surface energy balance, parameterizations of boundary-layer heights and the application of resistance laws near an Antarctic Ice Shelf front

A study of the surface energy balance with turbulent fluxes obtained by the Monin-Obukhov similarity theory and a comparison with results for resistance laws are presented for the strong baroclinic conditions in the vicinity of the Filchner/Ronne Ice Shelf front. The data are taken from a field experiment in the Antarctic summer season 1983/84. For the first time in the coastal Antarctic region, this data set comprises synchronous energy balance measurements over the polynya and the ice shelf together with soundings of the boundary layer, yielding vertical profiles of the wind velocity and temperature over the ice shelf, at the ice shelf front and over the polynya. Over the ice shelf, the radiation balance is the largest component of the energy fluxes and is mainly compensated by the subsurface energy flux and the turbulent heat flux in the daily mean. Over the polynya, turbulent fluxes of sensible and latent heat lead to large energy losses of the water surface in the night-time and in situations of very low air temperatures. Different parameterizations for boundary-layer height are compared using tethered sonde and energy balance measurements. With the height of the inversion base over the polynya and the height of the critical bulk Richardson number over the ice shelf, external parameters for the application of resistance laws were determined. The comparison of turbulent surface fluxes obtained by the energy balance measurements and by the resistance laws shows good agreement for the convective conditions over the polynya. For the stably stratified boundary layer over the ice shelf with small amounts of the turbulent heat flux, the deviation is large for the case of a cold air outflow with a superposed inertial oscillation.

Richardson number profiles in laboratory experiments applied to shallow seas

Abstract A unified analysis is given of the critical conditions for the onset of stratification due to either a vertical or a horizontal buoyancy flux, with tidal or wind stirring. The critical conditions for the onset of stratification with a horizontal buoyancy flux are found to be of the form of ratios of the tidal slope, or wind setup, to the equivalent surface slope due to the lateral density gradient. These ratios, which are easily determined from sea data, indicate that the profiles of critical flux Richardson Number, averaged over the stirring cycle, are similar to those inferred from the laboratory experiments of Hopfinger and Linden (1982) in which there is zero mean shear turbulence with a stabilising buoyancy flux, and also that the efficiency for the conversion of kinetic energy to potential energy for tidal stirring is similar to that for wind stirring. The observed much greater efficiency for wind stirring, compared with tidal stirring with a vertical buoyancy flux, is also consistent with ...

The stability of a compressible stratified shear layer

The stability of a stratified shear layer is studied using the compressible magnetohydrodynamic (MHD) equations, including an effective gravity term to represent curvature effects of the flow and magnetic field line geometry. A general eigenvalue equation is derived for a two-dimensional MHD fluid. For the case of a hyperbolic tangent shear flow and exponential density profile, it is found that in the Boussinesq approximation the compressibility raises the critical Richardson number from (1)/(4) to as much as (1)/(2) , with the exact value depending on the value of the magnetic field at infinity. Under the approximation of a strong asymptotic magnetic field, but without invoking the Boussinesq approximation, it is shown both analytically and numerically that the density gradient terms cause the shear instability to be dispersive. In particular, the long-wavelength stability boundary for the case of Richardson number J=0 is characterized by a normalized phase velocity c=−1. This result implies that the solution for the stability boundary found by Satyanarayana, Lee, and Huba [Phys. Fluids 30, 81 (1987)], which required that c=−β/2 (where β is the ratio of the velocity and density gradient scale lengths), is not valid, in general.

Numerical study of the Stratified smoke flow in a corridor: Full-scale calculations

The two-dimensional, steady-state, turbulent flow of hot air and smoke in a corridor with an adjacent room fire is calculated numerically. Turbulence is accounted for through the k-ϵ model. Buoyancy efects are included in both the vertical momentum and the turbulence equations. Convective and radiative heat losses are accounted for. The simple radiation model used assumes a well-mixed, spectrally gray and optically thick smoke layer. A full-sized corridor of 75 m length and 2.5 m height is considered. Parameter variations include inflow temperature, radiative heat losses and net mass flow. Special attention is given to the stratification of the hot smoke layer. Strong stratification is found for all situations considered. However, the hot layer thickness increases significantly in some cases. Our calculations do not support the use of a critical overall Richardson number as a criterion for the occurrence of stratification.

A Richardson Number criterion for the air-sea interface

It is shown that the observationally determined roughness relation z0 = αu*2/g in which g is the acceleration of gravity, u*, is the friction velocity in air, and α = 0.0185 (Wu, 1982) for the wind profile over the sea surface relative to the surface current, is consistent with the existence of a Richardson Number criterion at the air-sea interface in which the critical Richardson Number, Ric = 1, such that all the shear energy is converted into potential energy.

A parameterization of the stable atmospheric boundary layer

Two formulations of the stable atmospheric boundary layer are proposed for use in weather forecasting or climate models. They feature the log-linear profile near the surface, but are free from the associated critical Richardson number. The diffusion coefficients in the Ekman layer are a natural extension of the surface layer. They are locally determined using wind shear in one case and turbulent kinetic energy in the other. The parameterizations are tested in a one-dimensional model simulating the evolution of the nocturnal boundary layer with and without radiative cooling. Both formulations give very similar results, except near the top of the boundary layer where the transition to the free atmosphere is smoother with the wind shear formulation. A distinctive feature of these schemes is that they retain their simulating skill when resolution is reduced. This is verified for a wide range of situations. In practice, this means that there is no need for a large-scale model to have a level below 50 m or so.

A time‐dependent model for turbulent transfer in a stratified oceanic boundary layer

A first‐order model for vertical flux of momentum and scalars in a rotational boundary layer is applied to the oceanic boundary layer beneath sea ice. Model eddy viscosity is proportional to the product of the local friction velocity u* and a master length scale for vertical exchange, which is a function of the rotational length scale u*/ƒ and the local Obukhov length L. The ratio of eddy diffusivity to eddy viscosity is 1 when turbulence is energetic but falls to lower values when turbulence levels are low and stratification high, according to an empirical relation. There are three empirical constants in the theory: ξN, the ratio of the master length scale to the rotational length scale; Rc, the critical flux Richardson number; and a shape factor describing the falloff of the eddy diffusivity ratio at high gradient Richardson numbers. For an idealized simulation of melting and freezing conditions representative of the marginal ice zone, the model agrees closely with a similar implementation of the Mellor‐Yamada level 2.5 model. The model is demonstrated by performing a simulation of surface drift and mixed‐layer properties observed during the 1984 Marginal Ice Zone Experiment field study in the Greenland Sea.

Intense sheared flow as the origin of large‐scale undulations of the edge of the diffuse aurora

Using photographs from the DMSP satellite Lui et al. (1982) have found several examples of large scale undulations of the edge of the diffuse aurora. Several satellite experimenters have previously reported intense sheared flow near the edge of the diffuse aurora as evidenced either by detection of large localized meridional electric fields or by narrow regions of high azimuthal flow velocity. It is suggested here that the diffuse auroral undulations are caused by an instability of this sheared flow. In order to test this idea, a search for in situ satellite flow data associated with the events presented by Lui et al. was made. In addition the shear flow events reported by Rich et al. (1980) were used as a basis to search for near simultaneous DMSP photos. The cross‐section of these data sets was not large. The closest example was a satellite pass indicating shear 20 minutes after a DMSP photo showing strong undulations which had been developing for at least an hour. A theory recently published by Viñas and Madden (1986) is in good agreement with the observations in that, (1) several events studied by Rich et al. satisfied the magnetic Richardson number criterion for instability and (2) the observed wavelengths are in good agreement with the theory. A recent numerical simulation by Pritchett (1985) also seems to support the concept that the edge of the diffuse aurora may be unstable to a shear driven process. An interesting feature of the analysis presented here is that when scaled from ionospheric heights to the equatorial plane, the shear frequency is only reduced by 30–40% from ionospheric values.

Saturation and the “universal” spectrum for vertical profiles of horizontal scalar winds in the atmosphere

A theory is presented which explains the universal nature of one‐dimensional vertical wave number, k, power spectral densities (PSDs) of horizontal winds as measured in the atmosphere and predicted by VanZandt. The theory is that the PSD amplitude at any given wave number (greater than a certain minimum, k*) is determined by its saturation value due either to shear instability (i.e., critical Richardson Number) or, more likely, to convective instability. This explains why the PSD amplitudes observed do not grow exponentially with increasing altitude. This saturation theory assumption plus other considerations leads to a PSD of the form N2/kn, where n is in the range of about 2.5 to 3 and N is the Brunt frequency. A simplified model involving superimposed narrow bands of gravity waves as well as a model based merely on dimensional arguments both lead to n = 3. The full model not only explains the observed spectral slopes but also predicts the PSD amplitude in the troposphere to be 3.5 times smaller than in the stratosphere. The derivation of the model is based on the saturation condition that ∫ k2PSD(k) dk = N2. The model may also apply to the ocean and explain the Garrett‐Munk vertical wave number spectrum.

On the Energetics and Nonhydrostatic Aspects of Symmetric Baroclinic Instability

Dry symmetric baroclinic instability is studied using numerical analysis at the critical Richardson number (Ri) for varying thermal Rossby (Ro), Prandtl (Pr), and Ekman numbers. For small enough Ro, dissipation results in a lowering of the critical Ri as Ro decreases, and the critical wavelength is not proportional to Ro. For Pr = 1, the energy conversion is via both inertial and convective mechanisms.

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).

Richardson Number Criterion for the Nonlinear Stability of Three-Dimensional Stratified Flow

With use of a method of Arnol'd, we derive the necessary and sufficient conditions for the formal stability of a parallel shear flow in a three-dimensional stratified fluid. When the local Richardson number defined with respect to density variations is everywhere greater than unity, the equilibrium is formally stable under nonlinear pertrubations. The essential physical content of the nonlinear stability result is that the total energy acts as a potential for deformations of the fluid across constant density surfaces; this well is required to have definite curvature to assure stability under these deformations.

Three-dimensional baroclinic instability of a Hadley cell for small Richardson number

A three-dimensional linear stability analysis of a baroclinic flow for Richardson number Ri of order unity is presented. The model considered is a thin, horizontal, rotating fluid layer which is subjected to horizontal and vertical temperature gradients. The basic state is a Hadley cell which is a solution of the Navier–Stokes and energy equations and contains both Ekman and thermal boundary layers adjacent to the rigid boundaries; it is given in closed form. The stability analysis is also based on the Navier–Stokes and energy equations; and perturbations possessing zonal, meridional and vertical structures were considered. Numerical methods were developed for the solution of the stability problem, which results in an ordinary differential eigenvalue problem. The objectives of this work were to extend the previous theoretical work on three-dimensional baroclinic instability for small Ri to a more realistic model involving the Prandtl number σ and the Ekman number E, and to finite growth rates and a wider range of the zonal wavenumber. The study covers ranges of 0.135 [les ] Ri [les ] 1.1, 0.2 [les ] σ [les ] 5.0, and 2 × 10−4 [les ] E [les ] 2 σ 10−3. For the cases computed for E = 10−3 and σ ≠ 1, we found that conventional baroclinic instability dominates for Ri > 0.825 and symmetric baroclinic instability dominates for Ri < 0.675. However, for E [ges ] 5 × 10−4 and σ = 1 in the range 0.3 [les ] Ri [les ] 0.8, conventional baroclinic instability always dominates. Further, we found in general that the symmetric modes of maximum growth are not purely symmetric but have weak zonal structure. This means that the wavefronts are inclined at a small angle to the zonal direction. The results also show that as E decreases the zonal structure of the symmetric modes of maximum growth rate also decreases. We found that when zonal structure is permitted the critical Richardson number for marginal stability is increased, but by only a small amount above the value for pure symmetric instability. Because these modes do not substantially alter the results for pure symmetric baroclinic instability and because their zonal structure is weak, it is unlikely that they represent a new type of instability.

Symmetric Baroclinic Instability of a Hadley Cell

A symmetric baroclinic instability is examined in terms of a Boussinesq fluid contained between two horizontal plates to determine the effects of the Ekman and thermal layers. Governing equations are written for a rotating reference frame, taking into account the Rossby, Ekman, and Prandtl numbers. Equations are defined for the perturbation functions, treated as an eigenvalue problem, and a numerical integration of the full eighth order differential system is performed by a shooting technique. An instability is found to occur in the Hadley cell containing both Ekman and thermal boundary layers when the Richardson number is close to unity. If the Prandtl number is fixed the critical Richardson number decreases with an increasing Ekman number until the Ekman number reaches a certain value, at which time the fluid is stable.

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).

Efficiency of conversion of kinetic energy to potential energy by a breaking internal gravity wave

A simple model of turbulent mixing induced by a breaking internal gravity wave implies that one quarter of the kinetic energy lost goes into potential energy. The model is based on a critical Richardson number concept. The conceptual model and some numeric trials simulate the experiment of Thorpe (1973) quite well.

Decaying turbulence in neutral and stratified fluids

Decaying turbulence in neutral and stratified fluids has been studied experimentally for relatively high mesh Reynolds numbers and long time-histories. The neutral case indicates an initial period decay law, q2∝ t−1, through non-dimensional time \[ W_gt/M \simeq 800 \] which is considerably longer than previous measurements at the same mesh Reynolds number (Re = 48260). The stratified experiment resulted in a decay rate virtually identical to that of the neutral case through Wgt/M = 275. However the decay rate sharply decreased after this time when the field of turbulence was replaced by internal gravity waves. A critical Richardson number marks the transition from the turbulence to an internal gravity wave domain.

Atmospheric Sublayer Transport and Odor Control

Wastewater treatment plant boundary odor concentrations exceeding 5 ou/scf can cause odor complaints. Highest plant boundary odor concentrations for a given source odor emission rate occur when the stable sublayer temperature at 25 ft. (7.62m) minus that at 5 ft. (1.52m) exceeds 2°F (1.1°C). Another criterion for critical transport may be the occurrence of critical Richardson number conditions within the transporting sublayer. The effectiveness of dispersion-inducing barriers and wind machines is apparently reduced as the net heat flow from air to ground increases. Monitored micrometeorological variables are used to determine the magnitude and frequency of critical transport conditions. (Statistical procedures are used to obtain estimates of the frequency of occurrence of 5 ou/scf odor concentrations at the plant boundary from various odor sources with alternative control systems in use.) Limiting control system operation to critical transport conditions can reduce operating costs to a small fraction of continuous operation costs without incurring additional odor risks.