An improvement in clear-air turbulence forecasting based on spontaneous imbalance theory: the ULTURB algorithm

Recent research has shown that Lighthill–Ford spontaneous gravity wave generation theory, when applied to numerical model data, can help predict areas of clear-air turbulence. It is hypothesized that this is the case because spontaneously generated atmospheric gravity waves may initiate turbulence by locally modifying the stability and wind shear. As an improvement on the original research, this paper describes the creation of an ‘operational’ algorithm (ULTURB) with three modifications to the original method: (1) extending the altitude range for which the method is effective downward to the top of the boundary layer, (2) adding turbulent kinetic energy production from the environment to the locally produced turbulent kinetic energy production, and, (3) transforming turbulent kinetic energy dissipation to eddy dissipation rate, the turbulence metric becoming the worldwide ‘standard’. In a comparison of ULTURB with the original method and with the Graphical Turbulence Guidance second version (GTG2) automated procedure for forecasting mid- and upper-level aircraft turbulence ULTURB performed better for all turbulence intensities. Since ULTURB, unlike GTG2, is founded on a self-consistent dynamical theory, it may offer forecasters better insight into the causes of the clear-air turbulence and may ultimately enhance its predictability.

Turbulence dynamics near a turbulent/non-turbulent interface

The characteristics of the boundary layer separating a turbulence region from an irrotational (or non-turbulent) flow region are investigated using rapid distortion theory (RDT). The turbulence region is approximated as homogeneous and isotropic far away from the bounding turbulent/non-turbulent (T/NT) interface, which is assumed to remain approximately flat. Inviscid effects resulting from the continuity of the normal velocity and pressure at the interface, in addition to viscous effects resulting from the continuity of the tangential velocity and shear stress, are taken into account by considering a sudden insertion of the T/NT interface, in the absence of mean shear. Profiles of the velocity variances, turbulent kinetic energy (TKE), viscous dissipation rate (epsilon), turbulence length scales, and pressure statistics are derived, showing an excellent agreement with results from direct numerical simulations (DNS). Interestingly, the normalized inviscid flow statistics at the T/NT interface do not depend on the form of the assumed TKE spectrum. Outside the turbulent region, where the flow is irrotational (except inside a thin viscous boundary layer), epsilon decays as z^{-6}, where z is the distance from the T/NT interface. The mean pressure distribution is calculated using RDT, and exhibits a decrease towards the turbulence region due to the associated velocity fluctuations, consistent with the generation of a mean entrainment velocity. The vorticity variance and epsilon display large maxima at the T/NT interface due to the inviscid discontinuities of the tangential velocity variances existing there, and these maxima are quantitatively related to the thickness delta of the viscous boundary layer (VBL). For an equilibrium VBL, the RDT analysis suggests that delta ~ eta (where eta is the Kolmogorov microscale), which is consistent with the scaling law identified in a very recent DNS study for shear-free T/NT interfaces.

On the connection between dissipation enhancement in the ocean surface layer and Langmuir circulations

A mechanism for the enhancement of the viscous dissipation rate of turbulent kinetic energy (TKE) in the oceanic boundary layer (OBL) is proposed, based on insights gained from rapid-distortion theory (RDT). In this mechanism, which complements mechanisms purely based on wave breaking, preexisting TKE is amplified and subsequently dissipated by the joint action of a mean Eulerian wind-induced shear current and the Stokes drift of surface waves, the same elements thought to be responsible for the generation of Langmuir circulations. Assuming that the TKE dissipation rate epsilon saturates to its equilibrium value over a time of the order one eddy turnover time of the turbulence, a new scaling expression, dependent on the turbulent Langmuir number, is derived for epsilon. For reasonable values of the input parameters, the new expression predicts an increase of the dissipation rate near the surface by orders of magnitude compared with usual surface-layer scaling estimates, consistent with available OBL data. These results establish on firmer grounds a suspected connection between two central OBL phenomena: dissipation enhancement and Langmuir circulations.

A linear model for the structure of turbulence beneath surface water waves

The structure of turbulence in the ocean surface layer is investigated using a simplified semi-analytical model based on rapid-distortion theory. In this model, which is linear with respect to the turbulence, the flow comprises a mean Eulerian shear current, the Stokes drift of an irrotational surface wave, which accounts for the irreversible effect of the waves on the turbulence, and the turbulence itself, whose time evolution is calculated. By analysing the equations of motion used in the model, which are linearised versions of the Craik–Leibovich equations containing a ‘vortex force’, it is found that a flow including mean shear and a Stokes drift is formally equivalent to a flow including mean shear and rotation. In particular, Craik and Leibovich’s condition for the linear instability of the first kind of flow is equivalent to Bradshaw’s condition for the linear instability of the second. However, the present study goes beyond linear stability analyses by considering flow disturbances of finite amplitude, which allows calculating turbulence statistics and addressing cases where the linear stability is neutral. Results from the model show that the turbulence displays a structure with a continuous variation of the anisotropy and elongation, ranging from streaky structures, for distortion by shear only, to streamwise vortices resembling Langmuir circulations, for distortion by Stokes drift only. The TKE grows faster for distortion by a shear and a Stokes drift gradient with the same sign (a situation relevant to wind waves), but the turbulence is more isotropic in that case (which is linearly unstable to Langmuir circulations).

On the distortion of turbulence by a progressive surface wave

A rapid-distortion model is developed to investigate the interaction of weak turbulence with a monochromatic irrotational surface water wave. The model is applicable when the orbital velocity of the wave is larger than the turbulence intensity, and when the slope of the wave is sufficiently high that the straining of the turbulence by the wave dominates over the straining of the turbulence by itself. The turbulence suffers two distortions. Firstly, vorticity in the turbulence is modulated by the wave orbital motions, which leads to the streamwise Reynolds stress attaining maxima at the wave crests and minima at the wave troughs; the Reynolds stress normal to the free surface develops minima at the wave crests and maxima at the troughs. Secondly, over several wave cycles the Stokes drift associated with the wave tilts vertical vorticity into the horizontal direction, subsequently stretching it into elongated streamwise vortices, which come to dominate the flow. These results are shown to be strikingly different from turbulence distorted by a mean shear flow, when `streaky structures' of high and low streamwise velocity fluctuations develop. It is shown that, in the case of distortion by a mean shear flow, the tendency for the mean shear to produce streamwise vortices by distortion of the turbulent vorticity is largely cancelled by a distortion of the mean vorticity by the turbulent fluctuations. This latter process is absent in distortion by Stokes drift, since there is then no mean vorticity. The components of the Reynolds stress and the integral length scales computed from turbulence distorted by Stokes drift show the same behaviour as in the simulations of Langmuir turbulence reported by McWilliams, Sullivan & Moeng (1997). Hence we suggest that turbulent vorticity in the upper ocean, such as produced by breaking waves, may help to provide the initial seeds for Langmuir circulations, thereby complementing the shear-flow instability mechanism developed by Craik & Leibovich (1976). The tilting of the vertical vorticity into the horizontal by the Stokes drift tends also to produce a shear stress that does work against the mean straining associated with the wave orbital motions. The turbulent kinetic energy then increases at the expense of energy in the wave. Hence the wave decays. An expression for the wave attenuation rate is obtained by scaling the equation for the wave energy, and is found to be broadly consistent with available laboratory data.

Dissipation of shear-free turbulence near boundaries

The rapid-distortion model of Hunt & Graham (1978) for the initial distortion of turbulence by a flat boundary is extended to account fully for viscous processes. Two types of boundary are considered: a solid wall and a free surface. The model is shown to be formally valid provided two conditions are satisfied. The first condition is that time is short compared with the decorrelation time of the energy-containing eddies, so that nonlinear processes can be neglected. The second condition is that the viscous layer near the boundary, where tangential motions adjust to the boundary condition, is thin compared with the scales of the smallest eddies. The viscous layer can then be treated using thin-boundary-layer methods. Given these conditions, the distorted turbulence near the boundary is related to the undistorted turbulence, and thence profiles of turbulence dissipation rate near the two types of boundary are calculated and shown to agree extremely well with profiles obtained by Perot & Moin (1993) by direct numerical simulation. The dissipation rates are higher near a solid wall than in the bulk of the flow because the no-slip boundary condition leads to large velocity gradients across the viscous layer. In contrast, the weaker constraint of no stress at a free surface leads to the dissipation rate close to a free surface actually being smaller than in the bulk of the flow. This explains why tangential velocity fluctuations parallel to a free surface are so large. In addition we show that it is the adjustment of the large energy-containing eddies across the viscous layer that controls the dissipation rate, which explains why rapid-distortion theory can give quantitatively accurate values for the dissipation rate. We also find that the dissipation rate obtained from the model evaluated at the time when the model is expected to fail actually yields useful estimates of the dissipation obtained from the direct numerical simulation at times when the nonlinear processes are significant. We conclude that the main role of nonlinear processes is to arrest growth by linear processes of the viscous layer after about one large-eddy turnover time.

A global perspective on Langmuir turbulence in the ocean surface boundary layer.

The turbulent mixing in thin ocean surface boundary layers (OSBL), which occupy the upper 100 m or so of the ocean, control the exchange of heat and trace gases between the atmosphere and ocean. Here we show that current parameterizations of this turbulent mixing lead to systematic and substantial errors in the depth of the OSBL in global climate models, which then leads to biases in sea surface temperature. One reason, we argue, is that current parameterizations are missing key surface-wave processes that force Langmuir turbulence that deepens the OSBL more rapidly than steady wind forcing. Scaling arguments are presented to identify two dimensionless parameters that measure the importance of wave forcing against wind forcing, and against buoyancy forcing. A global perspective on the occurrence of waveforced turbulence is developed using re-analysis data to compute these parameters globally. The diagnostic study developed here suggests that turbulent energy available for mixing the OSBL is under-estimated without forcing by surface waves. Wave-forcing and hence Langmuir turbulence could be important over wide areas of the ocean and in all seasons in the Southern Ocean. We conclude that surfacewave- forced Langmuir turbulence is an important process in the OSBL that requires parameterization.

Bridging the Faraoni and Selli oceanic anoxic events: late Hauterivian to early Aptian dysaerobic to anaerobic phases in the Tethys.

A detailed geochemical analysis was performed on the upper part of the Maiolica Formation in the Breggia (southern Switzerland) and Capriolo sections (northern Italy). The analysed sediments consist of well-bedded, partly siliceous, pelagic carbonate, which lodges numerous thin, dark and organic-rich layers. Stable-isotope, phosphorus, organic-carbon and a suite of redox-sensitive trace-element contents (RSTE: Mo, U, Co, V and As) were measured. The RSTE pattern and Corg:Ptot ratios indicate that most organic-rich layers were deposited under dysaerobic rather than anaerobic conditions and that latter conditions were likely restricted to short intervals in the latest Hauterivian, the early Barremian and the pre-Selli early Aptian. Correlations are both possible with organic-rich intervals in central Italy (the Gorgo a Cerbara section) and the Boreal Lower Saxony Basin, as well as with the facies and drowning pattern in the Helvetic segment of the northern Tethyan carbonate platform. Our data and correlations suggest that the latest Hauterivian witnessed the progressive installation of dysaerobic conditions in the Tethys, which went along with the onset in sediment condensation, phosphogenesis and platform drowning on the northern Tethyan margin, and which culminated in the Faraoni anoxic episode. This episode is followed by further episodes of dysaerobic conditions in the Tethys and the Lower Saxony Basin, which became more frequent and progressively stronger in the late early Barremian. Platform drowning persisted and did not halt before the latest early Barremian. The late Barremian witnessed diminishing frequencies and intensities in dysaerobic conditions, which went along with the progressive installation of the Urgonian carbonate platform. Near the Barremian-Aptian boundary, the increasing density in dysaerobic episodes in the Tethyan and Lower Saxony Basins is paralleled by a change towards heterozoan carbonate production on the northern Tethyan shelf. The following return to more oxygenated conditions is correlated with the second phase of Urgonian platform growth and the period immediately preceding and corresponding to the Selli anoxic episode is characterised by renewed platform drowning and the change to heterozoan carbonate production. Changes towards more humid climate conditions were the likely cause for the repetitive installation of dys- to anaerobic conditions in the Tethyan and Boreal basins and the accompanying changes in the evolution of the carbonate platform towards heterozoan carbonate-producing ecosystems and platform drowning.

Progressive palaeoenvironmental change during the Late Barremian-Early Aptian as prelude to Oceanic Anoxic Event 1a: Evidence from the Gorgo a Cerbara section (Umbria-Marche basin, central Italy)

During Oceanic Anoxic Event 1a (OAE 1a, 120 Ma; Li et al., 2008), organic carbon-rich layers were deposited in marine environments under anoxic conditions on a global scale. In this study, palaeoenvironmental conditions leading to this event are characterised by studying the Upper Barremian to the Lower Aptian succession of the Gorgo a Cerbara section (central Italy). For this, an integrated multi-proxy approach (δ13Ccarb; δ13Corg; δ18O; phosphorus; Total Organic Carbon, TOC; bulk-rock mineralogy, as well as redox-sensitive trace elements — RSTEs) has been applied. During the LateBarremian, thin organic-rich layers occur episodically, and associated Corg:Ptot ratios indicate the presence of intermittent dysoxic to anoxic conditions. Coarse correlations are observed between TOC, P and biogenic silica contents, indicating links between P availability, productivity, and TOC preservation. However, the corresponding δ13Ccarb and δ18O records remain quite stable, indicating that these brief periods of enhanced TOC preservation did not have sufficient impact on the marine carbon reservoir to deviate δ13C records. Around the Barremian–Aptian boundary, TOC-enriched layers become more frequent. These layers correlate with negative excursions in the δ13Ccarb and δ13Corg records, possibly due to a warming period as indicated by the δ18O record. During the earliest Aptian, this warming trend is reverted into a cooling trend, which is then followed by an important warming step near the onset of Oceanic Anoxic Event 1a (OAE 1a). During this time period, organic-rich intervals occur, which are characterised by the progressive increase in RSTE. The warming step prior the onset of OAE 1a is associated with the well-known negative spike in δ13Ccarb and δ13Corg records, an important peak in P accumulation, RSTE enrichments and Corg:Ptot ratios indicating the prevalence of anoxic conditions. The Selli Level itself may document a cooling phase. RSTE enrichments and Corg:Ptot ratios confirm the importance of anoxic conditions during OAE 1a at this site. The Gorgo a Cerbara section is interpreted to reflect the progressive impact of palaeoenvironmental change related to the formation of the Ontong-Java plate-basalt plateau, which started already around the Barremian–Aptian boundary and culminated into OAE 1a.

Importance of oceanic resolution and mean state on the extra-tropical response to El Niño in a matrix of coupled models

The extra-tropical response to El Niño in configurations of a coupled model with increased horizontal resolution in the oceanic component is shown to be more realistic than in configurations with a low resolution oceanic component. This general conclusion is independent of the atmospheric resolution. Resolving small-scale processes in the ocean produces a more realistic oceanic mean state, with a reduced cold tongue bias, which in turn allows the atmospheric model component to be forced more realistically. A realistic atmospheric basic state is critical in order to represent Rossby wave propagation in response to El Niño, and hence the extra-tropical response to El Niño. Through the use of high and low resolution configurations of the forced atmospheric-only model component we show that, in isolation, atmospheric resolution does not significantly affect the simulation of the extra-tropical response to El Niño. It is demonstrated, through perturbations to the SST forcing of the atmospheric model component, that biases in the climatological SST field typical of coupled model configurations with low oceanic resolution can account for the erroneous atmospheric basic state seen in these coupled model configurations. These results highlight the importance of resolving small-scale oceanic processes in producing a realistic large-scale mean climate in coupled models, and suggest that it might may be possible to “squeeze out” valuable extra performance from coupled models through increases to oceanic resolution alone.

The expression of the end-Cenomanian oceanic anoxic event (OAE 2) in the Helvetic Alps

Upper Cenomanian pelagic sediments from the northern Alpine Helvetic fold-and-thrust belt (northern Tethyan margin) coeval with Oceanic Anoxic Event (OAE) 2 are characterized by the temporal persistence of micrite sedimentation and lack of organic carbon-rich layers. We studied an expanded section in the Chrummflueschlucht (east of Euthal, Switzerland), which encompasses the OAE 2 time interval. In order to identify the paleoceanographic and paleoenvironmental conditions during OAE 2 in this part of the northern Tethyan margin, and more specifically to trace eventual changes in nutrient levels and oxic conditions, we investigated the biostratigraphy (planktonic foraminifera), the bulk-rock mineralogy, and measured stable carbon- and oxygen-isotopes, total phosphorus (P) and redox-sensitive trace-element (RSTE) contents. We were able to determine – with some remaining uncertainties – the different planktonic foraminiferal biozones characteristic of the Cenomanian–Turonian boundary interval (Rotalipora cushmani, Whiteinella archaeocretacea and Helvetoglobotruncana helvetica zones). In the lower part of the section (R. cushmani total range zone), the bulk-rock δ13C record shows a long-term increase. Within sediments attributed to the W. archaeocretacea partial range zone, δ13C values reach a maximum of 3.3‰ (peak “a”). In the following the values decrease and increase again to arrive at a plateau with high δ13C values of around 3.1‰, which ends with a peak of 3.3‰ (peak “c”). At the top of the section, in sediments belonging to the H. helvetica total range zone, δ13C values decrease to post-OAE values of around 2.2‰. The last occurrence of R. cushmani is observed just above the positive δ13C shift characterizing OAE 2. P contents display small variations along the section with a long-term decreasing trend towards the top. Before the OAE 2 interval, P values show higher values and relatively good covariation with detrital input, indicating higher nutrient input before OAE 2. In sediments corresponding to the onset of the δ13C positive excursion, P content is marked by a sharp peak probably linked to a slowdown in sedimentation rates and/or the presence of a small hiatus, as is shown by the presence of glauconite and phosphatic grains. In the interval corresponding to OAE 2, P values remain low and increase slightly at the end of the positive shift in the δ13C record (in the H. helvetica total range zone). The average contents of RSTE (U, V, As, Co, Mo and Mn) remain low throughout the section and appreciable RSTE enrichments have not been observed for the sedimentary interval corresponding to OAE 2. No correlation is observed with stratigraphic trends in RSTE contents in organic-rich deeper-water sections. The presence of double-keeled planktonic foraminifera species during most of the Cenomanian/Turonian boundary event is another evidence of relatively well-oxygenated conditions in this part of the northern Tethyan outer shelf. Our results show that the Chrummflueschlucht section corresponds to one of the most complete section for the Cenomanian–Turonian boundary interval known from the Helvetic realm even if a small hiatus may be present at the onset of the δ13C record (peak “a”). The evolution of P contents suggests an increase in input of this nutritive element at the onset of OAE2. However, the trends in RSTE contents and the planktonic foraminifera assemblages show that the Helvetic realm has not been affected by strongly depleted oxygen conditions during OAE 2.

The Valanginian δ13C excursion may not be an expression of a global oceanic anoxic event

Marine and terrestrial sediments of the Valanginian age display a distinct positive δ13C excursion, which has recently been interpreted as the expression of an oceanic anoxic episode (OAE) of global importance. Here we evaluate the extent of anaerobic conditions in marine bottom waters and explore the mechanisms involved in changing carbon storage on a global scale during this time interval. We asses redox-sensitive trace-element distributions (RSTE; uranium, vanadium, cobalt, arsenic and molybdenum) and the quality and quantity of preserved organic matter (OM) in representative sections along a shelf-basin transect in the western Tethys, in the Polish Basin and on Shatsky Rise. OM-rich layers corresponding in time to the δ13C shift are generally rare in the Tethyan sections and if present, the layers are not thicker than several centimetres and total organic carbon (TOC) contents do not surpass 1.68 wt..%. Palynological observations and geochemical properties of the preserved OM suggest a mixed marine and terrestrial origin and deposition in an oxic environment. In the Polish Basin, OM-rich layers show evidence for an important continental component. RSTE exhibit no major enrichments during the δ13C excursion in all studied Tethyan sections. RSTE enrichments are, however, observed in the pre-δ13C excursion OM-rich “Barrande” levels of the Vocontian Trough. In addition, all Tethyan sections record higher Mn contents during the δ13C shift, indicating rather well-oxygenated bottom waters in the western Tethys and the presence of anoxic basins elsewhere, such as the restricted basins of the North Atlantic and Weddell Sea. We propose that the Valanginian δ13C shift is the consequence of a combination of increased OM storage in marginal seas and on continents (as a sink of 12C-enriched organic carbon), coupled with the demise of shallow-water carbonate platforms (diminishing the storage capacity of 13C-enriched carbonate carbon). As such the Valanginian provides a more faithful natural analogue to present-day environmental change than most other Mesozoic OAEs, which are characterized by the development of ocean-wide dysaerobic to anaerobic conditions.

Extensivity of two-dimensional turbulence

This study is concerned with how the attractor dimension of the two-dimensional Navier–Stokes equations depends on characteristic length scales, including the system integral length scale, the forcing length scale, and the dissipation length scale. Upper bounds on the attractor dimension derived by Constantin, Foias and Temam are analysed. It is shown that the optimal attractor-dimension estimate grows linearly with the domain area (suggestive of extensive chaos), for a sufficiently large domain, if the kinematic viscosity and the amplitude and length scale of the forcing are held fixed. For sufficiently small domain area, a slightly “super-extensive” estimate becomes optimal. In the extensive regime, the attractor-dimension estimate is given by the ratio of the domain area to the square of the dissipation length scale defined, on physical grounds, in terms of the average rate of shear. This dissipation length scale (which is not necessarily the scale at which the energy or enstrophy dissipation takes place) can be identified with the dimension correlation length scale, the square of which is interpreted, according to the concept of extensive chaos, as the area of a subsystem with one degree of freedom. Furthermore, these length scales can be identified with a “minimum length scale” of the flow, which is rigorously deduced from the concept of determining nodes.

Constraints on the spectral distribution of energy and enstrophy dissipation in forced two-dimensional turbulence

We study two-dimensional (2D) turbulence in a doubly periodic domain driven by a monoscale-like forcing and damped by various dissipation mechanisms of the form νμ(−Δ)μ. By “monoscale-like” we mean that the forcing is applied over a finite range of wavenumbers kmin≤k≤kmax, and that the ratio of enstrophy injection η≥0 to energy injection ε≥0 is bounded by kmin2ε≤η≤kmax2ε. Such a forcing is frequently considered in theoretical and numerical studies of 2D turbulence. It is shown that for μ≥0 the asymptotic behaviour satisfies ∥u∥12≤kmax2∥u∥2, where ∥u∥2 and ∥u∥12 are the energy and enstrophy, respectively. If the condition of monoscale-like forcing holds only in a time-mean sense, then the inequality holds in the time mean. It is also shown that for Navier–Stokes turbulence (μ=1), the time-mean enstrophy dissipation rate is bounded from above by 2ν1kmax2. These results place strong constraints on the spectral distribution of energy and enstrophy and of their dissipation, and thereby on the existence of energy and enstrophy cascades, in such systems. In particular, the classical dual cascade picture is shown to be invalid for forced 2D Navier–Stokes turbulence (μ=1) when it is forced in this manner. Inclusion of Ekman drag (μ=0) along with molecular viscosity permits a dual cascade, but is incompatible with the log-modified −3 power law for the energy spectrum in the enstrophy-cascading inertial range. In order to achieve the latter, it is necessary to invoke an inverse viscosity (μ<0). These constraints on permissible power laws apply for any spectrally localized forcing, not just for monoscale-like forcing.

An ‘ideal’ form of decaying two-dimensional turbulence

In decaying two-dimensional Navier-Stokes turbulence, Batchelor's similarity hypothesis fails due to the existence of coherent vortices. However, it is shown that decaying two-dimensional turbulence governed by the Harney-Hasegawa-Mima (CHM) equation ∂/∂t (V^2 φ-λ^2 φ)+J(φ,∇^2 φ)=D where D is a damping, is described well by Batchelor's similarity hypothesis for wave numbers k ≪ λ (the so-called AM regime). It is argued that CHM turbulence in the AM regime is a more `ideal' form of two-dimensional turbulence than is Navier-Stokes turbulence itself.