An experimental study of nonlinear baroclinic instability and mode selection in a large basin

This paper reports on two-layer rotating liquid experiments designed to study the behavior of non-linear baroclinic waves under conditions where the Rossby radius of deformation R_d is much smaller than the geometric length scale L imposed by the size of the laboratory apparatus. The apparatus is constructed to consistently simulate f-plane dynamics. When F = L²/R_d² > > 1, it is found that the unstable waves first encountered as friction is decreased have high frequencies, in accord with linear theory. As the friction parameter $\text{Q = 0.7 E}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{/R}{}_0$ (where E is the Ekman number and R₀ the Rossby number) is further decreased into the non-linear region, singlewave amplitude vacillation is observed. Generally, as Q decreases lower frequencies (and low wavenumbers) dominate the response, which ultimately becomes turbulent at values of Q of the order 0.1. This is contrary to the result expected from an extrapolation of linear theory. Further observations show that the finite-amplitude state is not unique: multi-equilibria are possible depending on the initial conditions.     

A parameter study of the mixed instability of idealized ocean currents

We investigate the nature of linear instabilities that can arise on eastward-flowing baroclinic currents similar to those found to serve as sites of strong eddy-mean flow interaction in certain mesoscale-resolution ocean circulation studies. The intent is to deduce the dependence of the linear instability mechanism — thought to be operative in some form in these simulations — on the internal parameters characterizing them. Following conventional practice, we adopt as our physical model the two-level quasigeostrophic potential vorticity equations which, in their linearized form, are solved numerically to yield the properties of the most unstable linear waves under a variety of mean flow and environmental conditions. The kinematic and dynamic features of the growing perturbations — preferred wavelength, growth rate and frequency, eddy-mean field energy transfers and vertical distribution of wave amplitude — are shown to be sensitive functions of our nondimensional parameters: $\text{(}\text{i}\text{) α = (}\text{U}{}_3\text{U}{}_1\text{)}$, the ratio of lower to upper level velocity scale amplitude; $\text{(}\text{ii}\text{) X = (}\text{R}{}_{\mathrm{<mtext>d</mtext>}}\text{L}\text{)}$, the ratio of the first baroclinic deformation radius to the meridional width of the jet; $\text{(}\text{iii}\text{) δ = (}\text{H}{}_1\text{H}{}_3\text{)}$, the resting layer depth ratio; and $\text{(}\text{iv}\text{) ? = (}\text{βL}{}^2\text{U}\text{)}$, an (inverse) Rossby number based on the northward gradient of the planetary vorticity (β). Viscous effects, although included in the analysis, are shown to be unimportant for values of frictional coefficients typical of recent eddy-resolving ocean model studies. Despite a strong dependence of the details of the linear instability mechanism on environmental factors, the associated unstable eigenmodes do have important structural similarities which are intimately connected with their ability to extract energy from the mean flow.     

Experiments with viscous source flows in rotating systems

The dynamics of oceanic bottom currents are examined both theoretically and in the laboratory. A class of similarity solutions for steady flow indicates that the geostrophic current is drained by Ekman flux at its downslope edge and ultimately extinguished at a downstream distance of order $\text{(}\text{fQ}\text{s}{}^2\text{g}{}_{\mathrm{r}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ magnified by $\text{E}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$.Laboratory source flows are found to be consistently wave-like. Nevertheless, certain gross features of the steady Ekman flux mechanism are observed. Instabilities are classified according to the magnitudes of Rossby (?) and Ekman (E) numbers for the steady flow scaling. Characteristic were forms include: (1) a meandering jet (E < ?, 10^?2); (2) transverse waves on broad contour current (? < E, 10^?2); and (3) transverse waves on a viscous flow (?, E > 10^?2). The dispersion relation for source flow waves resembles that of baroclinic instabilities for a uniform two-layer channel flow, and an empirically determined stability boundary is in rough agreement with the inviscid channel flow criterion.An interpretation of field measurements from the Denmark Strait Overflow in terms of the laboratory results is presented.     

The effect of ekman suction on a flow over a shallow topography in the beta plane

The effect of bottom Ekman layer suction on a homogeneous, constant depth, eastwards, low Rossby number flow over a shallow bottom topography in the beta plane is studied. The governing vorticity equation is obtained by expanding the velocities in the continuity and momentum equations in powers of the Rossby number, ?, and matching the vertical velocity with the vertical velocity at the outer edge of the bottom Ekman layer obtained from the Ekman layer solution. The suction effect is then linearized using an Oseen approxiamation and the resulting linear model is solved using Fourier transforms with the requirement that the solution behave like a vortex near the origin which is equivalent to the effect of an isolated bump, i.e., a Green's function solution is obtained. An analytical solution is thus, obtained in integral form and then numerically integrated. The effect of Ekman suction is found to be a damping of the downstream Rossby waves in a distance of order $\text{2√2U/f}{}_0\text{E}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, an increased upstream influence, and a counterclockwise rotation of the closed streamline region about the origin. It is pointed out that the vortex solutions can be superimposed in order to obtain the solution for flow over topographies of finite horizontal text. This technique was used to compute the flow over a right circular cylinder. The results agree favorably with the experimental results of McCartney (1975).     

Selective withdrawal of rotating stratified fluid

A simple theory and some experimental observations are presented of the transient withdrawal of rotating, stratified fluid in a field of gravity. The problem is confined to axisymmetric geometry and negligible viscosity. It is predicted that the withdrawal initially proceeds like non-rotating selective withdrawal, but at a time equal to $\text{3√}\text{3}\text{2?}$ there is a transition to a rotation-dominated selective withdrawal process which requires that fluid come from distances above and/or below the inlet given by the time-dependent formula $\text{(?Qt/2πr}{}_0\text{N)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. Experimental observations are given which are in approximate agreement with the predictions.     

Mixing efficiency in decaying stably stratified turbulence

Mixing efficiency in stratified flows is a measure of the proportion of turbulent kinetic energy that goes into increasing the potential energy of the fluid by irreversible mixing. In this research direct numerical simulations (DNS) and rapid distortion theory (RDT) calculations of transient turbulent mixing events are carried out in order to study this aspect of mixing. In particular, DNS and RDT of decaying, homogeneous, stably-stratified turbulence are used to determine the mixing efficiency as a function of the initial turbulence Richardson number R_it0=(N_L0/_u0)²$R{i}_{t0}={(N{L}_0/u_0)}^2$, where N   is the buoyancy frequency and _L0$L_0$ and _u0$u_0$ are initial length and velocity scales of the turbulence. The results show that the mixing efficiency increases with increasing R_it0$R{i}_{t0}$ for small R_it0$R{i}_{t0}$, but for larger R_it0$R{i}_{t0}$ the mixing efficiency becomes approximately constant. These results are compared with data from towed grid experiments. There is qualitative agreement between the DNS results and the available experimental data, but significant quantitative discrepancies. The grid turbulence experiments suggest a maximum mixing efficiency (at large R_it0$R{i}_{t0}$) of about 6%, while the DNS and RDT results give about 30%. We consider two possible reasons for this discrepancy: Prandtl number effects and non-matching initial conditions. We conclude that the main source of the disagreement probably is due to inaccuracy in determining the initial turbulence energy input in the case of the grid turbulence experiments.     

Markov chain simulations of vertical dispersion from elevated sources into the neutral planetary boundary layer

A Lagrangian statistical-trajectory model based on a Markov chain relation is used to investigate vertical dispersion from elevated sources into the neutral planetary boundary layer. The model is fully two-dimensional, in that both vertical and longitudinal velocity fluctuations, and their correlation, are simulated explicitly. The best observational information currently available is used to characterize the mean and turbulent structure of the neutral boundary layer. In particular, a realistic vertical profile of the Lagrangian integral time scale is proposed, based partly on a review of direct measurements and partly on a comparison of the model predictions with published diffusion data. The model predictions are shown to agree well with a variety of dispersion observations. The model is used to study vertical diffusion as a function of release height H, friction velocity u* and surface roughness z ₀ for downwind distances up to 10 km from the source. The equivalent Gaussian dispersion parameter Σ _z is shown to decrease slightly with an increase in H, and to increase with increases in z ₀ or u*. It is demonstrated that relationships valid in a field of homogeneous turbulence can be applied to vertical dispersion in the atmosphere if the release occurs above the region of strongest gradients in the mean and turbulent parameters. Scaling in terms of the standard deviation in elevation angle of the wind at the release point leads to a universal curve which provides accurate estimates of Σ _z over a wide range of values of H, z ₀ and the meteorological parameters.     

Surface influence upon vertical profiles in the nocturnal boundary layer

Near-surface wind profiles in the nocturnal boundary layer, depth h, above relatively flat, tree-covered terrain are described in the context of the analysis of Garratt (1980) for the unstable atmospheric boundary layer. The observations at two sites imply a surface-based transition layer, of depth z _*, within which the observed non-dimensional profiles Φ _M ⁰ are a modified form of the inertial sub-layer relation \(\Phi _M \left( {{z \mathord{\left/ {\vphantom {z L}} \right. \kern-0em} L}} \right) = \left( {{{1 + 5_Z } \mathord{\left/ {\vphantom {{1 + 5_Z } L}} \right. \kern-0em} L}} \right)\) according to $$\Phi _M^{\text{0}} \simeq \left( {{{1 + 5z} \mathord{\left/ {\vphantom {{1 + 5z} L}} \right. \kern-\nulldelimiterspace} L}} \right)\exp \left[ { - 0.7\left( {{{1 - z} \mathord{\left/ {\vphantom {{1 - z} z}} \right. \kern-\nulldelimiterspace} z}_ * } \right)} \right]$$ , where z is height above the zero-plane displacement and L is the Monin-Obukhov length. At both sites the depth z _* is significantly smaller than the appropriate neutral value (z _*N ) found from the previous analysis, as might be expected in the presence of a buoyant sink for turbulent kinetic energy.     

True gravity in ocean dynamics Part 1 Ekman transport

The direction normal to the Earth spherical (or ellipsoidal) surface is not vertical (called deflected vertical) since the vertical direction is along the true gravity g (= ig_λ+ jg_φ+ kg_z). Here, (λ, φ, z) are (longitude, latitude, depth), and (i, j, k) are the corresponding unit vectors. The spherical (or ellipsoidal) surfaces are not horizontal surfaces (called deflected-horizontal surfaces). The most important body force g (true gravity) has been greatly simplified without justification in oceanography to the standard gravity (-g₀k) with g₀ = 9.81 m/s². Impact of such simplification on ocean dynamics is investigated in this paper using the Ekman layer model. In the classical Ekman layer dynamic equation, the standard gravity (-g₀k) is replaced by the true gravity g(λ, φ, z) with a constant eddy viscosity and a depth-dependent-only density ρ(z) represented by an e-folding near-inertial buoyancy frequency. New Ekman spiral and in turn new formulae for the Ekman transport are obtained for ocean with and without bottom. With the gravity data from the global static gravity model EIGEN-6C4 and the surface wind stress data from the Comprehensive Ocean-Atmosphere Data Set (COADS), large difference is found in the Ekman transport using the true gravity and standard gravity.     

Measurements of spray at an air-water interface

The spray content in the surface boundary layer above an air—water interface was determined by a series of measurements at various feteches and wind speeds in a laboratory facility. The droplet flux density N(z) can be described in terms of the scaling flux density N_* and von Karman constant K throguh the equation, N(z)/N_* = −(1/K) ln(z/z_0d) where z is height above the mean water level and z_0d is the droplet boundary layer thickness. N_* is given by a unique relationship in terms of the roughness Reynolds number u_*σ/ν where σ is the root-mean-square surface displacement. Spray inception occurred for u_* 0.3. The dominant mode of spray generation in the present and most other laboratory tests, as well as in available field data, appears to be bubble bursting.     

Periodic solutions of the set of equations governing the nonadiabatic convection of dry isolated thermals

The changes with timet of a temperature deviation δT(t,α) and of a vertical velocityW _i(t,α) of an isolated dry thermal have been investigated theoretically. Solutions for the functionW _i(t, α) have been derived for stable and unstable environmental stratifications. Comparing these solutions with the corresponding ones for the rise of an adiabatic thermal yield some interesting conclusions. Firstly, there is the evident relation between the rate of entrainment of environmental air (expressed by the parameter α=(1/M _i) dM _i/dz whereM _i is the mass of the thermal) and the vertical velocity of the thermal: an increase in α decreases the velocity. Two similar thermals in stably stratified surroundings, one of them moving adiabatically (α=0) the other nonadiabatically (α>0), would rise for the same length of timet ₂=π/N, whereN is a typical Brunt-Väisälä frequency, but with different velocities and to different heights: the ascent timet ₂ depends only on environmental parameters. In an unstable stratification, the vertical non-adiabatic velocity of the thermal, instead of increasing without limit, tends towards a finite asymptotic velocity $$W_t (\infty ) = (\sqrt { - \mathcal{N}^2 } )/\alpha $$ the value of which depends upon both the stratification of the surroundings and upon the entrainment rate α. In a real atmosphere, where additional retarding forces exist, the motion will certainly be damped.     

Heat and Mass Transfer to and from Surfaces with Dense Vegetation or Similar Permeable Roughness

A differential equation is obtained to describe the concentration of passive admixtures (water vapor, sensible heat, pollutants, CO₂, etc.) of turbulent flow inside a dense and uniform vegetational canopy. The profiles of eddy diffusivity, wind speed and shear stress are assumed to be exponential decay functions of depth below the top of the canopy. This equation is solved for the case of a vegetation with constant concentration of the admixture at the foliage surfaces. The solution is used to formulate bulk mass or heat transfer coefficients, which can be applied to practical problems involving surfaces covered with a vegetation or with similar porous or fibrous roughness elements. The results are shown to be consistent with experimental data presented by Chamberlain (1966), Garratt and Hicks (1973) and Garratt (1978). Calculations with the model illustrate that, as compared to its behavior over surfaces with bluff roughness elements, ln(z ₀/ z _oc ) (where z ₀ is the momentum roughness and Z _oc the scalar roughness) for permeable roughness elements is relatively insensitive to u _* and practically independent of z ₀.     

Structures of dynamically unstable shear flows and their implications for shallow internal gravity waves

Summary In this study, the response of a dynamically unstable shear flow with a critical level to periodic forcing is presented. An energy argument is proposed to explain the upshear tilt of updrafts associated with disturbances in two-dimensional stably stratified flows. In a dynamically unstable flow, the energy equation requires an upshear tilt of the perturbation streamfunction and vertical velocity whereU _z is positive. A stability model is constructed using an iteration method. An upshear tilt of the vertical velocity and the streamfunction fields is evident in a dynamically unstable flow, which is required by energy conversion from the basic shear to the growing perturbation wave energy according to the energy argument. The momentum flux profile indicates that the basic flow is decreased (increased) above (below) the critical level. Thus, the shear instability tends to smooth the shear layer. Following the energy argument, a downshear tilt of the updraft is produced in an unstably stratified flow since the perturbation wave energy is negative. The wave energy budget indicates that the disturbance is caused by a thermal instability modified by a shear flow since the potential energy grows faster than the kinetic energy.With 4 Figures     

Aerodynamic Roughness Length of Fresh Snow

This study presents the results from a series of wind-tunnel experiments designed to investigate the aerodynamic roughness length z ₀ of fresh snow under no-drift conditions. A two-component hot-film anemometer was employed to obtain vertical profiles of velocity statistics in a zero pressure gradient turbulent boundary layer for flow over naturally deposited snow surfaces. The roughness of these snow surfaces was measured by means of digital photography to capture characteristic length scales that can be related to z ₀. Our results show that, under aerodynamically rough conditions, the mean value of the roughness length for fresh snow is \({\langle{z}_{0}\rangle= 0.24}\) mm with a standard deviation σ(z ₀) = 0.05 mm. In this study, we show that variations in z ₀ are associated with variations in the roughness geometry. The roughness measurements suggest that the estimated values of z ₀ are consistent with the presence of irregular roughness structures that develop during snowfalls that mimic ballistic deposition processes.