Rossby Similarity Research Articles

Rossby number similarity of an atmospheric RANS model using limited-length-scale turbulence closures extended to unstable stratification

Abstract. The design of wind turbines and wind farms can be improved by increasing the accuracy of the inflow models representing the atmospheric boundary layer. In this work we employ one-dimensional Reynolds-averaged Navier–Stokes (RANS) simulations of the idealized atmospheric boundary layer (ABL), using turbulence closures with a length-scale limiter. These models can represent the mean effects of surface roughness, Coriolis force, limited ABL depth, and neutral and stable atmospheric conditions using four input parameters: the roughness length, the Coriolis parameter, a maximum turbulence length, and the geostrophic wind speed. We find a new model-based Rossby similarity, which reduces the four input parameters to two Rossby numbers with different length scales. In addition, we extend the limited-length-scale turbulence models to treat the mean effect of unstable stratification in steady-state simulations. The original and extended turbulence models are compared with historical measurements of meteorological quantities and profiles of the atmospheric boundary layer for different atmospheric stabilities.

One‐dimensional evolution of the upper water column in the Atlantic sector of the Arctic Ocean in winter

AbstractA one‐dimensional model is employed to reproduce the observed time evolution of hydrographic properties in the upper water column during winter, between 26 January and 11 March 2015, in a region north of Svalbard in the Nansen Basin of the Arctic Ocean. From an observed initial state, vertical diffusion equations for temperature and salinity give the hydrographic conditions at a later stage. Observations of microstructure are used to synthesize profiles of vertical diffusivity, K, representative of varying wind forcing conditions. The ice‐ocean heat and salt fluxes at the ice‐ocean interface are implemented as external source terms, estimated from the salt and enthalpy budgets, using friction velocity from the Rossby similarity drag relation, and the ice core temperature profiles. We are able to reproduce the temporal evolution of hydrography satisfactorily for two pairs of measured profiles, suggesting that the vertical processes dominated the observed changes. Sensitivity tests reveal a significant dependence on K. Variation in other variables, such as the temperature gradient of the sea ice, the fraction of heat going to ice melt, and the turbulent exchange coefficient for heat, are relatively less important. The increase in salinity as a result of freezing and brine release is approximately 10%, significantly less than that due to entrainment (90%) from beneath the mixed layer. Entrainment was elevated during episodic storm events, leading to melting. The results highlight the contribution of storms to mixing in the upper Arctic Ocean and its impact on ice melt and mixed‐layer salt and nutrient budgets.

Effects of a Shallow Pycnocline and Surface Meltwater on Sea Ice–Ocean Drag and Turbulent Heat Flux

Abstract Comprehensive boundary layer measurements from a drift station on first-year ice in the late summer of 2012 in the Nansen basin, when stable stratification in the upper ocean extended all the way to the surface, are analyzed. Observed quadratic ice–ocean drag coefficients, based on measurements of wind stress, are roughly 3.6 × 10−3, consistent with neutral-stability Rossby similarity scaling. The turning angles of 32°–39° between surface velocity and stress are larger than Rossby similarity predicts and obey a different scaling. This can be explained by the shallow pycnocline forcing the Ekman transport into a thin layer and modeled roughly employing a simple first-order correction to Rossby similarity. Turbulent shear stress in the ice–ocean boundary layer is on average 3 times smaller than the estimate based on wind stress, possibly because internal wave drag was significant. This lowers vertical scalar fluxes by 38% compared to a scenario where turbulent stress accounts for the total drag. The authors measure an average upward ocean–ice heat flux of 10 W m−2, which is 50% smaller than predicted by a bulk heat flux parameterization. This reduction is attributed to additional sources of heat and freshwater that alter the ice–ocean interface salt balance. This study shows that a commonly used bulk heat flux parameterization is a special case of a simple downgradient parameterization allowing for a modified interface salt budget. For similar wind forcing, observed ice–ocean fluxes of heat and salt were 40%–100% larger when the ice-relative current approached from a nearby pressure ridge keel than otherwise.

Ekman Veering, Internal Waves, and Turbulence Observed under Arctic Sea Ice

Abstract The ice–ocean system is investigated on inertial to monthly time scales using winter 2009–10 observations from the first ice-tethered profiler (ITP) equipped with a velocity sensor (ITP-V). Fluctuations in surface winds, ice velocity, and ocean velocity at 7-m depth were correlated. Observed ocean velocity was primarily directed to the right of the ice velocity and spiraled clockwise while decaying with depth through the mixed layer. Inertial and tidal motions of the ice and in the underlying ocean were observed throughout the record. Just below the ice–ocean interface, direct estimates of the turbulent vertical heat, salt, and momentum fluxes and the turbulent dissipation rate were obtained. Periods of elevated internal wave activity were associated with changes to the turbulent heat and salt fluxes as well as stratification primarily within the mixed layer. Turbulent heat and salt fluxes were correlated particularly when the mixed layer was closest to the freezing temperature. Momentum flux is adequately related to velocity shear using a constant ice–ocean drag coefficient, mixing length based on the planetary and geometric scales, or Rossby similarity theory. Ekman viscosity described velocity shear over the mixed layer. The ice–ocean drag coefficient was elevated for certain directions of the ice–ocean shear, implying an ice topography that was characterized by linear ridges. Mixing length was best estimated using the wavenumber of the beginning of the inertial subrange or a variable drag coefficient. Analyses of this and future ITP-V datasets will advance understanding of ice–ocean interactions and their parameterizations in numerical models.

Study of tributary inflows in Lake Iseo with a rotating physical model

The influence of Coriolis force on the currents of large lakes is well acknowledged; very few contributions, however, investigate this aspect in medium-size lakes where its relevance could be questionable. In order to study the area of influence of the two major tributary rivers in Lake Iseo, a rotating vertically distorted physical model of the northern part of this lake was prepared and used, respecting both Froude and Rossby similarity. The model has a horizontal length scale factor of 8000 and a vertical scale factor of 500 and was used both in homogeneous and in thermally stratified conditions. We explored the pattern of water circulation in front of the entrance mouth for different hydrologic scenarios at the beginning of spring and in summer. We neglected the influence of winds. The primary purposes of the model were twofold: i) to increase our level of knowledge of the hydrodynamics of Lake Iseo by verifying the occurrence of dynamical effects related to the Earth’s rotation on the plume of the two tributaries that enter the northern part of the lake and ii) to identify the areas of the lake that can be directly influenced by the tributaries’ waters, in order to provide guidance on water quality monitoring in zones of relevant environmental and touristic value. The results of the physical model confirm the relevant role played by the Coriolis force in the northern part of the lake. Under ordinary flow conditions, the model shows a systematic deflection of the inflowing waters towards the western shore of the lake. The inflow triggers a clockwise gyre within the Lovere bay, to the West of the inflow, and a slow counter-clockwise gyre, to the East of the inflow, that returns water towards the river mouth along the eastern shore. For discharges with higher return period, when only the contribution by Oglio River is relevant, the effect of the Earth’s rotation weakens in the entrance zone and the plume has a more rectilinear pattern, whilst in the far field the current driven by the inflows keeps moving along the western shore. On the basis of these results one could expect that the north-western part of the lake between Castro and Lovere, although not aligned with the tributaries’ axes, is more sensitive to accumulation effects related to river-borne pollution. The results obtained with the physical model are critically compared with data obtained from different sources: the trajectory of a lagrangian drogue; a map of reflectivity data from the lake floor; a map of water turbidity at the intrusion depth. The findings are also confirmed by the results of a 3D numerical model of the lake .

Advances in understanding ice–ocean stress during and since AIDJEX

Abstract Variation of ice/ocean drag (momentum exchange) is an important yet often overlooked aspect of pack ice modeling. It is commonly parameterized as proportional to the square of the velocity difference between the ice and the undisturbed ocean, often with a constant angle offset to account for rotational effects in the ice–ocean boundary layer. This approach is critiqued in light of extensive observations that have revealed the underlying turbulence scales governing momentum exchange within the IOBL. Fluid dynamical similarity implied by these scales provides a framework for addressing several factors that affect the drag relationship, including variation in ice roughness, relative drift speed, buoyancy flux at the ice/ocean interface, and stratification in the upper ocean. These are examined and discussed in light of recent changes in the Arctic ice pack. The drag law is formulated in terms of dimensionless surface velocity, which in its simplest form is called Rossby similarity, and accounts explicitly for variation in undersurface hydraulic roughness, z 0 . A generalization that includes interfacial buoyancy flux is also described and illustrated, and the impact of near surface ocean stratification is discussed. Estimates of z 0 based on underice measurements vary widely; by a combination of observations and simple IOBL modeling, an attempt is made to reduce these to a manageable set associated with distinct ice types.

Ocean Heat Flux in the Central Weddell Sea during Winter

Abstract Seasonal sea ice, which plays a pivotal role in air–sea interaction in the Weddell Sea (a region of large deep-water formation with potential impact on climate), depends critically on heat flux from the deep ocean. During the austral winter of 1994, an intensive process-oriented field program named the Antarctic Zone Flux Experiment measured upper-ocean turbulent fluxes during two short manned ice-drift station experiments near the Maud Rise seamount region of the Weddell Sea. Unmanned data buoys left at the site of the first manned drift provided a season-long time series of ice motion, mixed layer temperature and salinity, plus a (truncated) high-resolution record of temperature within the ice column. Direct turbulence flux measurements made in the ocean boundary layer during the manned drift stations were extended to the ice–ocean interface with a “mixing length” model and were used to evaluate parameters in bulk expressions for interfacial stress (a “Rossby similarity” drag law) and ocean-to-...

Physical Model of Niagara River Discharge

Development of a physical model study for the Niagara River discharge into Lake Ontario is described. The model is designed on the basis of (distorted) Froude and Rossby similarity and the results demonstrate clearly the effects of rotation on the flow. Important features of the plume are shown to be reproduced by the model, including its anticyclonic (right-turning) flow and the subsequent formation of a near-shore current along the southern coastline. A scaling analysis of the integrated longitudinal momentum equation is suggested as a framework for analyzing this type of flow. This analysis considers three regions of the flow: (1) a near-field, where inertial forces are balanced by pressure and bottom drag; (2) an intermediate-field where the balance is between inertia and buoyancy; and (3) afar-field characterized by nearly geostrophic flow (balance between buoyancy and rotation). This scheme is then tested against conditions characteristic of the Niagara River discharge during spring flows and also with observations from the physical model tests, showing reasonable agreement. One particular result of interest is that the distance at which rotation starts to turn the flow scales approximately with the internal Rossby radius. Results of the study help to distinguish between the relative effects of buoyancy and rotation.

Aircraft estimates of the geostrophic drag coefficient and the rossby similarity functions A and B over the sea

Estimates of the geostrophic drag coefficient and the Rossby similarity functions, A and B obtained from data collected by an instrumented aircraft over the sea are presented. The average value of the geostrophic drag coefficient is 0.027 and is independent of the geostrophic windspeed. The dependence of the similarity functions A and B on boundary-layer parameters is investigated. The function A is found to depend on baroclinicity parameters, while B depends on the parameter u*/fh (where u* is the surface friction velocity, f is the Coriolis parameter, and h is the boundary-layer depth). Using the geostrophic drag coefficient found here and the results of surface drag coefficient studies, a relationship between geostrophic windspeed and surface windspeed is obtained which shows good agreement with empirical data.

Aircraft estimates of the geostrophic drag coefficient and the Rossby similarity functions A and B over the sea

Aircraft estimates of the geostrophic drag coefficient and the Rossby similarity functions A and B over the sea

Rossby similarity and turbulent formulations

The structure of the Planetary Boundary Layer in the stationary case is investigated. The dynamic equations are written in an universal form deduced from Rossby Similarity. Then the system is closed using two semi-empirical formulations for the turbulent fluxes (Prandtl's formulation and the turbulent energy scheme) with a mixing-length chosen to be compatible with Rossby Similarity. For each formulation the system is solved using a new numerical procedure to compute universal profiles of wind and also the Similarity functions A and B in terms of stability. These profiles and functions are then compared to experimental data and good agreement is obtained. Investigating simpler formulations, it appears that good results are obtained with a simple two-layer model, patching a diabatic Ekman spiral with surface-layer profiles. Its formulation is analytical and it can incorporate constant baroclinicity.

海上大気境界層に於ける顕熱と水蒸気輸送に対するロスビー相似則の係数

AMTEX観測に於いて得られた海上のラジオゾンデのデータを用いて,大気境界層のロスビー相似則の係数C及びDの値を評価した。大気境界層の高さhは0.3u*/ |f|に等しいと仮定した。温度と温度の鉛直分布について,海面直上に接する層流副層が考慮されている。得られた係数Cは中立温度成層に近い場合,不安定成層ならば正の大きな値,また,安定成層ならば負の大きな値をとる。従来の研究では,Cは中立に近い条件下で安定度と共にゆるやかに変化するとされているが,本研究の結果はこのような従来の研究結果と一見矛盾するように見える。しかし詳しく調べてみると,本研究の結果は合理的であり,従来の結果とも本質的には矛盾していないことが理解される。本研究で得られた係数C,Dの実験公式を用いて,ロスビー相似則に基いて計算した顕熱と水蒸気輸送量の値はパルク法によるものとよく一致する。

Similarity parameters from first-order closure and data

A two-layer, first-order closure model for the Planetary Boundary Layer (PBL) is developed with the objective of parameterizing the surface stress with respect to the synoptic scale. The model includes stability effects by considering stratification-dependent secondary flow in the outer layer and empirical corrections to the surface layer flow. It shows the compatibility of simple eddy viscosity closure solutions with similarity theory by producing the now well-known Rossby similarity equations. It allows further insight into the Rossby similarity parameters by relating them to a single similarity parameter which is the ratio of the characteristic scales of the PBL and the surface layer. The measured and derived values of the similarity parameters A and B are compared with AIDJEX data and other published values. The variation in these values in stably stratified conditions is predicted and two alternate similarity parameters are calculated, one a constant and the other with a small variation and decreasing influence on the drag coefficient in stable stratification. The result is an empirical resistance law for a geostrophic drag coefficient variation which parameterizes an observed order-of-magnitude change in surface stress with changes in roughness or PBL stratification. This variation is related to similarity parameters characteristic of the region and to measurable changes in the geostrophic departure angle.