Importance of subgrid-scale parameterization in numerical simulations of lake circulation

Two subgrid-scale modeling techniques––Smagorinsky’s postulation for the horizontal eddy viscosity and the Mellor–Yamada level-2 model for the vertical eddy viscosity––are applied as turbulence closure conditions to numerical simulations of resolved-scale baroclinic lake circulations. The use of the total variation diminishing (TVD) technique in the numerical treatment of the advection terms in the governing equations depresses numerical diffusion to an acceptably low level and makes stable numerical performances possible with small eddy viscosities resulting from the turbulence closure parameterizations. The results show that, with regard to the effect of an external wind stress, the vertical turbulent mixing is mainly restricted to the topmost epilimnion with the order of magnitude for the vertical eddy viscosity of 10⁻³ m² s⁻¹, whilst the horizontal turbulent mixing may reach a somewhat deeper zone with an order of magnitude for the horizontal eddy viscosity of 0.1–1 m² s⁻¹. Their spatial and temporal variations and influences on numerical results are significant. A comparison with prescribed constant eddy viscosities clearly shows the importance of subgrid-scale closures on resolved-scale flows in the lake circulation simulation. A predetermination of the eddy viscosities is inappropriate and should be abandoned. Their values must be determined by suitable subgrid-scale closure techniques.     

Turbulence flux measurement above the overstory of a subtropical Pinus plantation over the hilly region in southeastern China

Continuous turbulence flux measurement using the eddy covariance (EC) technique was made from January 1 to December 31 in 2003 at two and three canopy heights of a subtropical Pinus plantation on the red earth hilly region in southeastern China. To be able to make sure that the measured turbulence flux will equal the net ecosystem/atmosphere exchange, the quality of the data has to be assessed. Three criteria were investigated here, including the power spectra and cospectra analyses, flux variance similarity (integral turbulence test) and energy balance closure. The spectral analyses suggested that above-canopy power spectral slopes for all velocity components and scalars such as CO2, H2O and air temperature followed the expected -2/3 power law in the inertial subrange, and their cospectral slopes were close to -4/3 power law in the inertial subrange. The important contribution of large-scale motions to energy and mass transfer above the canopy at higher measurement level was also confirmed by the spectral analyses. The eddy covariance systems have the ability to resolve fluctuations associated with small-scale eddies and did not induce an obvious underestimation of the measured turbulence flux. The Monin-Obukhov similarity functions for the normalized standard deviation of vertical wind speed and air temperature were well-defined functions of atmospheric stability at two heights above the forest canopy, which indicated that turbulence flux measurements made at two heights were within the surface layer. Nocturnal flux underestimation and departures of this normalized standard deviation of vertical wind speed similarity function from that expected from Monin-Obukhov theory were a function of friction velocity. Thus, an optimal criterion of friction velocity was determined to be greater than 0.2-0.3 m s-1 for nocturnal fluxes so that the eddy covariance flux measurement was under high turbulent mixing conditions. Energy balance closure reached about 72%-81% at the studied site, which was comparable to the 10%-30% of energy imbalance reported in the literature. However, the energy balance closure could be only used as a useful reference criterion.     

近地层能量闭合度对陆面过程模式影响

大量近地层观测试验表明,利用涡动相关法观测的湍流通量小于近地层可利用能量,即近地层能量是不闭合的,这种不闭合度一般为20%甚至更高.而陆面过程模式是基于地气间能量平衡建立,并且模式中的湍流边界层参数化方案通常根据实际观测的湍流通量来确定,因此能量不闭合必将对陆面过程模式造成一定的影响.本文利用2007年春季SACOL站的近地层观测资料,依据能量守恒将能量不闭合中的残余能量通过波文比分配到观测的湍流通量中,即修正涡动相关法观测的湍流通量使得近地层能量达到平衡;之后分别利用观测和修正的湍流通量,建立了能量不闭合和闭合情形下的湍流参数化方案,借助陆面过程模式SHAW,通过数值模拟和对比分析方法考察近地层能量闭合度对陆面过程模式的影响.研究结果表明近地层能量闭合对陆面过程模式有显著的影响:在陆面过程数值模拟中,当应用近地层能量不闭合的湍流通量形成的湍流参数化方案时,陆面过程模式会明显高估地表长波辐射及土壤温度;但当应用修正湍流通量使得近地层能量达到闭合形成的湍流参数化方案后,在不改变任何地表土壤物理生化属性的情况下,陆面过程模式能较好地模拟地表长波辐射和土壤温度.     

An analysis of the Chow-Kulandaiswamy GHS model

The Chow-Kulandaiswamy general hydrologic system (GHS) model is revisited. Based on a mathematical study by Singh and McCann the GHS model is simplified. Explicit solutions are obtained for special cases which can satisfactorily determine watershed surface runoff response due to given rainfall excess. A rational criterion is developed to determine the number of derivative terms to be retained in the model. In order to determine the coefficients in the GHS model the method of moments is proposed. Criteria are developed to determine complex roots and oscillations for these coefficients. By analysing Chow-Kulandaiswamy's results it is found that in a majority of cases which they studied roots are complex. Moreover, for the cases which have complex roots, a majority of the solutions oscillate. A brief sensitivity analysis of the GHS model is performed with regard to: (a) its leading coefficient, and (b) the order of the differential equation. Finally, the peak characteristics are specified for the second order case and their qualitative properties are shown for the third order case.     

Wind and melt driven circulation in a marginal sea ice edge frontal system: a numerical model

A seasonal ice edge zone is a unique frontal system with an air-ice-sea interface. This paper is a report on the numerical results from a quasi-three dimensional, time dependent, non-linear numerical model of circulation at a continental shelf-seasonal ice edge zone. The purpose of the experiments is to model the hydrography and circulation, including upwelling, baroclinic geostrophic flow, and inertial oscillations, at the ice edge with emphasis on examining the driving forces of wind and melting ice. It is suggested that the non-linear acceleration terms and vertical density diffusion terms are negligible and that the horizontal density diffusion terms are of secondary importance within the time and space scales of the experiments. The vertical eddy viscosity terms are important in a spin-up time scale and for Ekman transport and a bottom Ekman layer. The effects of the horizontal eddy viscosity terms are observable (a long-ice jet is diffused away from the ice edge) by the end (72 h) of the model runs.Model results are compared with available oceanographic and meteorological data for verification. The observed and modeled features of melt water induced water column stability, frontal structure, and ice edge upwelling are briefly discussed relative to observed ice edge primary production. Because the model is relatively general in nature, it is readily applicable to other seasonal or marginal ice edge zones in either hemisphere.     

Benchmarking the mesoscale variability in global ocean eddy-permitting numerical systems

The role of data assimilation procedures on representing ocean mesoscale variability is assessed by applying eddy statistics to a state-of-the-art global ocean reanalysis (C-GLORS), a free global ocean simulation (performed with the NEMO system) and an observation-based dataset (ARMOR3D) used as an independent benchmark. Numerical results are computed on a 1/4 ^° horizontal grid (ORCA025) and share the same resolution with ARMOR3D dataset. This “eddy-permitting” resolution is sufficient to allow ocean eddies to form. Further to assessing the eddy statistics from three different datasets, a global three-dimensional eddy detection system is implemented in order to bypass the need of regional-dependent definition of thresholds, typical of commonly adopted eddy detection algorithms. It thus provides full three-dimensional eddy statistics segmenting vertical profiles from local rotational velocities. This criterion is crucial for discerning real eddies from transient surface noise that inevitably affects any two-dimensional algorithm. Data assimilation enhances and corrects mesoscale variability on a wide range of features that cannot be well reproduced otherwise. The free simulation fairly reproduces eddies emerging from western boundary currents and deep baroclinic instabilities, while underestimates shallower vortexes that populate the full basin. The ocean reanalysis recovers most of the missing turbulence, shown by satellite products , that is not generated by the model itself and consistently projects surface variability deep into the water column. The comparison with the statistically reconstructed vertical profiles from ARMOR3D show that ocean data assimilation is able to embed variability into the model dynamics, constraining eddies with in situ and altimetry observation and generating them consistently with local environment.     

A three-dimensional numerical shelf-sea front model with variable eddy viscosity and diffusivity

A three-dimensional numerical model of circulation and eddy development in shelf-sea fronts is applied to three frontal structures, with two parameterization schemes for vertical eddy viscosity and diffusivity. The three fronts resemble those in the German Bight (a front between relatively fresh coastal water and saltier water offshore, with an interface extending from surface to bottom), the Norwegian Coastal Current (also formed by fresh coastal water but with a thermocline on one side), and the Celtic Sea (a front between water which is stratified in summer and water which is well mixed throughout the year). The two mixing assumptions, modelling the reduction of turbulence in stratified zones, are based on the Munk-Anderson scheme and the turbulent energy equation. Many features of frontal dynamics are common to all the results: strong surface currents along the front, cross-frontal circulation cells, a considerable enhancement of vertical velocities when eddies are formed, and development of eddies into cyclonic-anticyclonic vortex pairs. Cross-frontal circulation and frontal sharpening are the variables most sensitive to the different mixing assumptions. The German Bight front is the one most affected by changing these assumptions. The comparisons suggest that realistic results may be obtained from models despite the present uncertainty about vertical mixing in stratified shelf seas.     

Three-dimensional structure of mesoscale eddies in the western tropical Pacific as revealed by a high-resolution ocean simulation

The three-dimensional structure of mesoscale eddies in the western tropical Pacific(6°S–20°N, 120°E–150°E)is investigated using a high-resolution ocean model simulation. Eddy detection and eddy tracking algorithms are applied to simulated horizontal velocity vectors, and the anticyclonic and cyclonic eddies identified are composited to obtain their three-dimensional structures. The mean lifetime of all long-lived eddies is about 52 days, and their mean diameter is 147 km. Two typical characteristics of mesoscale eddies are revealed and possible dynamic explanations are analyzed. One typical characteristic is that surface eddies are generally separated from subthermocline eddies along the bifurcation latitude(~13°N) of the North Equatorial Current in the western tropical Pacific, which may be associated with different eddy energy sources and vertical eddy energy fluxes in subtropical and tropical gyres. Surface eddies have maximum swirl velocities of 8–9 cm s~(-1) and can extend to about 1500 m depth. Subthermocline eddies occur below 200 m, with their cores at about 400–600 m depth, and their maximum swirl velocities can reach 10 cm s~(-1). The other typical characteristic is that the meridional velocity component of the eddy is much larger than the zonal component. This characteristic might be due to more zonal eddy pairs(two eddies at the same latitude),which is also supported by the zonal wavelength(about 200 km) in the high-frequency meridional velocity component of the horizontal velocity.     

Properties of QBO and SAO generated by gravity waves

In this paper we present an extension for the 2D (zonal mean) version of our numerical spectral mode (NSM) that incorporates Hines’ Doppler spread parameterization (DSP) for small-scale gravity waves (GW). This model is applied to describe the seasonal variations and the semi-annual and quasi-biennial oscillations (SAO and QBO). Our earlier model reproduced the salient features of the mean zonal circulation in the middle atmosphere, including the QBO extension into the upper mesosphere inferred from UARS measurements. The model is extended to reproduce the upwelling at equatorial latitudes that is associated with the Brewer–Dobson circulation — which affects significantly the dynamics of the stratosphere as Dunkerton had pointed out. In the presence of GW, this upwelling is produced in our model with tropospheric heating, which generates also zonal jets outside the tropics similar to those observed. The resulting upward vertical winds increase the period of the QBO. To compensate for that, one needs to increase the eddy diffusivity and the GW momentum flux, bringing the latter closer to values recommended in the DSP. The QBO period in the model is 30 months (mo), which is conducive to synchronize this oscillation with the seasonal cycle of solar forcing. Associated with this QBO are interannual and interseasonal variations that become increasingly more important at higher altitudes — and this variability is interpreted in terms of GW filtering that effectively couples the dynamical components of the mesosphere. The computed temperature amplitudes for the SAO and QBO are in substantial agreement with observations at equatorial and extra-tropical latitudes. At high latitudes, however, the observed QBO amplitudes are significantly larger, which may be a signature of propagating planetary waves not included in the present model. The assumption of hydrostatic equilibrium not being imposed, we find that the effects from the vertical Coriolis force associated with the equatorial oscillations are large for the vertical winds and significant for the temperature variations even outside the tropics, but the effects are small for the zonal winds.     

A model of the wave and current boundary‐layer structure

A model of the wave and current boundary‐layer structure was developed using the k–ε turbulent closure model. The finite‐difference method was used to solve the governing equations. Vertical logarithmic grids and equal time steps were adopted. The following modelled simulations were obtained: (1) vertical profiles of wave velocity amplitude, eddy viscosity coefficient and turbulent kinetic energy with waves only; (2) vertical profiles of wave velocity amplitude, mean current velocity, eddy viscosity coefficient and turbulent kinetic energy with waves having a following current. To test the validity and the rationality of the present model, vertical profiles of modelled wave velocity amplitude and mean velocity were compared with corresponding experimental results available in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical model studies on the generation and dissipation of the diurnal coastal-trapped waves over the Sakhalin shelf in the Sea of Okhotsk

To clarify the generation and dissipation mechanisms of diurnal coastal-trapped waves (CTWs) over the Sakhalin shelf, a series of numerical experiments were conducted using a three-dimensional tidal model of the Okhotsk Sea with density stratification. The tidal model used has good reproduction owing to the careful fitting to the recent observations. The numerical experiments suggested that diurnal CTWs are primarily (~60%) generated by the conversion of tidal energy at the northern corner of the Sakhalin shelf, and further amplified by vorticity generation due to the water column oscillation from Sakhalin Bay and the influence of Kashevarov Bank. From the observations, it was found that diurnal CTWs are effectively dissipated by the strong spin-down due to bottom friction. The conventional turbulent closure model cannot reproduce the observed damping of diurnal CTWs, which raises a caution in modeling the tidal fields in high-latitude regions where diurnal CTWs exist. To resolve this underestimation of the damping, the vertical eddy viscosity was parameterized using its dependence on the observed major axis length of the diurnal tidal current ellipses, which improves the model reproduction on the damping of diurnal CTWs. The model also suggests that the spin-down effects due to friction associated with the sea-ice cover play an important role in the tidal current reduction in the region where diurnal CTWs exist, as the observations suggested.     

Experimental and numerical modelling of sedimentation in a rectangular shallow basin

Numerical simulation of flows in shallow reservoirs has to be checked for its consistency in predicting real flow conditions and sedimentation patterns. Typical flow patterns may exhibit flow separation at the inlet, accompanied by several recirculation and stagnation areas all over the reservoir surface. The aim of the present research project is to study the influence of the geometry of a reservoir on sediment transport and deposition numerically and experimentally, focusing on a prototype reservoir depth between 5 and 15 m as well as suspended sediment transport.
A series of numerical simulations is presented and compared with scaled laboratory experiments, with the objective of testing the sensitivity to different flow and sediment parameters and different turbulence closure schemes. Different scenarios are analyzed and a detailed comparison of preliminary laboratory tests and some selected simulations are presented.
The laboratory experiments show that suspended sediment transport and deposition are determined by the initial flow pattern and by the upstream and downstream boundary conditions. In the experiments, deposition in the rectangular basin systematically developed along the left bank, although inflow and outflow were positioned symmetrically along the centre of the basin. Three major horizontal eddies developed influencing the sediment deposition pattern. Although asymmetric flow patterns are privileged, a symmetric pattern can appear from time to time. This particular behaviour could also be reproduced by a two-dimensional depth-averaged flow and sediment transport model （CCHE2D）. The paper presents numerical simulations using different turbulence closure schemes （k-ε and eddy viscosity models）. In spite of the symmetric setup, these generally produced an asymmetric flow pattern that can easily switch sides depending on the assumptions made for the initial and boundary conditions. When using the laboratory experiment as a reference, the most reliable numerical results have been obtai     

Oceanic upwelling and downwelling processes in waters off the west coast of India

A three-dimensional model based on the Princeton Ocean Model (POM) has been implemented to study the circulation of the west coast of India. The model uses a curvilinear orthogonal horizontal grid with higher resolution near the coast (3–9 km) and a terrain following sigma coordinate in the vertical. The model is able to simulate Lakshadweep High and Lakshadweep Low (LL) during the winter and summer monsoons, respectively. During winter, the downwelling processes noticed along the coast help in the formation of temperature inversions. The inversions can be seen even up to the depths of ～50 m, which agrees with the available ARGO data in the region. Model simulations show that coastal upwelling off Kerala is at its peak in July. The intensity of upwelling reduces along the coast towards north. During the existence of LL, there is a cyclonic eddy in the sub-surface waters over the South-Eastern Arabian Sea, with vertical extent up to the depths of 100–150 m and it is strengthened due to the presence of northward counter current in the shelf region. The southerly coastal jet formed along the southern coast as a result of upwelling is noticed a westward shift along with LL. The location of the eddy off Kerala is tilted towards the open ocean with depth and our experiments suggest that this flow can be understood as a first baroclinic mode.     

Turbulence and mixing in a freshwater-influenced tidal bay: Observations and numerical modeling

In situ observations and numerical simulations of turbulence are essential to understanding vertical mixing processes and their dynamical controls on both physical and biogeochemical processes in coastal embayments. Using in situ data collected by bottom-mounted acoustic Doppler current profilers(ADCPs) and a free-falling microstructure profiler, as well as numerical simulations with a second-moment turbulence closure model, we studied turbulence and mixing in the Xiamen Bay, a freshwater-influenced tidal bay located at the west coast of the Taiwan Strait. Dynamically, the bay is driven predominantly by the M2 tide, and it is under a significant influence of the freshwater discharged from the Jiulong River. It is found that turbulence quantities such as the production and dissipation rates of the turbulent kinetic energy(TKE) were all subject to significant tidal variations, with a pronounced ebb-flood asymmetry. Turbulence was stronger during flood than ebb. During the flooding period, the whole water column was nearly well mixed with the depth-averaged TKE production rate and vertical eddy viscosity being up to 5?10?6 W kg?1 and 2?10?2 m2 s?1, respectively. In contrast, during the ebb strong turbulence was confined only to a 5?8 m thick bottom boundary layer, where turbulence intensity generally decreases with distance from the seafloor. Diagnosis of the potential energy anomaly showed that the ebb-flood asymmetry in turbulent dissipation and mixing was due mainly to tidal straining process as a result of the interaction between vertically shared tidal currents and horizontal density gradients. The role of vertical mixing in generating the asymmetry was secondary. A direct comparison of the modeled and observed turbulence quantities confirmed the applicability of the second-moment turbulence closure scheme in modeling turbulent processes in this weakly stratified tidally energetic environment, but also pointed out the necessity of further refinements of the model.     

An Analysis of Turbulent Heat Fluxes and the Energy Balance During the REFLEX Campaign

Three eddy covariance stations were installed at the Barrax experimental farm during the Land-Atmosphere Exchanges (REFLEX) airborne training and measurement campaign to provide ground truth data of energy balance fluxes and vertical temperature and wind profiles. The energy balance closure ratio (EBR) was 105% for a homogeneous camelina site, 86% at a sparse reforestation site, and 73% for a vineyard. We hypothesize that the lower closure in the last site was related to the limited fetch. Incorporating a vertical gradient of soil thermal properties decreased the RMSE of the energy balance at the camelina site by 16 W m^?2. At the camelina site, eddy covariance estimates of sensible and latent heat fluxes could be reproduced well using mean vertical profiles of wind and temperature, provided that the Monin—Obukhov length is known. Measured surface temperature and sensible heat fluxes suggested high excess resistance for heat (kB^?1 = 17).     

Simulating the temporal and spatial dynamics of the North Sea using the new model GETM (general estuarine transport model)

Here we present results of a 1-year realistic North Sea simulation from the new model GETM (general estuarine transport model) and assess the capabilities of this model by comparing them to model results from the well-known HAMSOM (Hamburg shelf sea and ocean model) model, in situ data from the North Sea project and satellite-derived sea-surface temperature data. The annual cycle and the spatial variability of stratification and mixing in the North Sea is simulated. It is shown that the new model is successful in reproducing the general temporal and spatial dynamics of the North Sea. The major advantages of GETM for achieving improved results in this simulation are the implementation of general vertical coordinates, of a state-of-the-art turbulence model and of higher-order advection schemes. By exploiting the full capabilities of these features a more realistic simulation could be achieved. We found that the greatest differences in the model results are produced by applying advection schemes of different complexity. Here we are able to demonstrate that better advection schemes lead to stronger horizontal gradients and stronger vertical stratification during summer. When comparing these results to measurements from the North Sea project and to satellite data, we find that these stronger gradients are more realistic. Therefore, we consider it as essential to use such high-order advection schemes if the spatial variability of estuarine or shelf seas like the North Sea is to be resolved adequately. The advanced turbulence closure scheme also contributed to more realistic simulation of the vertical stratification. Finally, general vertical coordinates better resolve the shallow regions, but are also useful for the deeper regions, as they allow a better estimation of sea-surface temperature compared to traditional coordinates.Responsible Editor: Phil Dyke     

A three-dimensional model study of near-inertial motion: generation and influence of an along-shelf flow

A three-dimensional baroclinic model of the Balearic Sea region is used to examine the processes influencing the distribution of near-inertial currents and waves in the region. Motion is induced by a spatially uniform wind impulse. By using a uniform wind, Ekman pumping due to spatial variability in the wind is removed with the associated generation of internal waves. However, internal waves can still be produced where stratification intersects topography. The generation and propagation of these waves, together with the spatial distribution of wind-forced inertial oscillations, are examined in detail. Diagnostic calculations show that in the near-coastal region inertial oscillations are inhibited by the coastal boundary. Away from this boundary the magnitude of the inertial oscillations increases, with currents showing a 180° phase shift in the vertical. The inclusion of an along-shelf flow modifies the inertial currents due to non-linear interaction between vorticity in the flow and the inertial oscillations. Prognostic calculations show that besides inertial oscillations internal waves are generated. In a linear model the addition of an along-shelf flow produces a slight reduction in the energy at the near-inertial frequency due to enhanced viscosity associated with the flow and changes in density field. The inclusion of non-linear effects modifies the currents due to inertial oscillations in a manner similar to that found in the diagnostic model. A change in the effective inertial frequency also influences the propagation of the internal waves. However, this does not appear to be the main reason for the enhanced damping of inertial energy, which is due to the along-shelf advection of water of a different density into a region and increased viscosity and mixing associated with the along-shelf flow.Responsible Editor: Phil Dyke     

Behaviour of a hydrodynamic finite element model

A finite element model which solves the vertically integrated momentum and continuity equations is described. Linear triangular elements are used to describe the geometry and parameter variations. The Galerkin method of weighted residuals is employed to cast the equations in a form amenable to numerical solution. The model is based on a fully implicit formulation using finite differences for the temporal derivatives.Means of evaluating the non-linear terms of the governing equations are described, and model results are presented for a frictionless tidal channel. The example is chosen such that the non-linearities have a large influence on the solution, and as a result the linearization scheme significantly affects the model's behaviour.Suppression of the non-linear instabilities generated by the convective terms in the momentum equations is examined for the case of flow around a 180° bend. Both the imposition of artificially high roughness coefficients and the use of an effective eddy viscosity are examined in terms of their ability to damp the oscillations which arise for this example.Finally, model results are presented for a case study involving determination of remedial measures to improve flow conditions at a river outfall in Southern Ontario.     

Operational Advanced Air Pollution Modeling

— Operational models that use solutions of the advection-diffusion equation based on more realistic assumptions than that of homogeneous wind and eddy diffusivity coefficients are presented. In particular a new parameterization for a model using a solution that accepts wind and eddy diffusivity profiles described by power functions of height is introduced. The performance of the model with the new parameterization was assessed using experimental data sets.