SCIENTIFIC PROGRAMS AND ACTIVITIES

October 24, 2014

July 27-29, 2009
Workshop on the Dynamics in Environmental and Geophysical Flows
at the University of Waterloo

Organizers: Marek Stastna, Francis Poulin

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Invited Speaker Abstracts

Leon Boegman
Queen's University

Flow separation and resuspension beneath shoaling nonlinear internal waves

Laboratory observations are presented showing the structure and dynamics of the turbulent bottom boundary layer beneath nonlinear internal waves (NLIWs) of depression shoaling upon sloping topography. The adverse pressure gradient beneath the shoaling waves causes the rear face to steepen, flow separation to occur, and wave-induced near-bottom vortices to suspend bed material. The resuspension is directly attributed to the near-bed viscous stress and to near-bed patches of elevated positive Reynolds stress generated by the vortical structures. These results are consistent with published field observations of resuspension events beneath shoaling NLIWs. Elevated near-bed viscous stresses are found throughout the domain at locations that are not correlated to the resuspension events. Near-bed viscous stress is thus required for incipient sediment motion but is not necessarily a precursor for resuspension. Resuspension is dependent on the vertical velocity field associated with positive Reynolds stress and is also found to occur where the mean (wave-averaged) vertical velocity is directed away from the bed. The results are interpreted by analogy to the eddy-stress and turbulent bursting
resuspension models developed for turbulent channel flows.

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Lydia Bourouiba
York University

Role of nonlocal interactions in the two-dimensionalization of forced rotating turbulence

Rotating flows are characterized by the nondimensional Rossby number Ro=U/Lf, where U is the characteristic velocity, L the characteristic length scale, and f is the Coriolis parameter. When rotation is increased and Ro goes to 0, the nonlinearity of the equations of motion becomes weak and the theories of weak wave interactions apply. The normal modes of the flow can be decomposed into zero-frequency 2D large scale structures and inertial waves (3D). Using numerical simulations of forced rotating turbulence with a combination of forcing schemes and resolutions, we find a robust regime separation with three regimes analagous to those identified in decaying rotating turbulence (Bourouiba and Bartello 2007). The study of discreteness effects on resonant and near-resonant mode interactions showed the robustness of the regime separation free of discreteness effects (Bourouiba 2008). The strong two-dimensionalization in the interemdiate Ro regime is characterized by a 2D energy spectrum of slope -3 or steeper, independent of the forcing scheme. We discuss the nonlocal or local nature of the dominant interactions involved, with a particular focus on the interactions between the 2D and the 3D modes. We find that the dominant interactions characterizing the intermediate Ro regime and leading to the steep slope for the 2D energy spectrum in forced flows are nonlocal and associated with a condensate of 2D energy in the large scales.

Coauthors: D. Straub (McGill University)

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Magda Carr
University of St Andrews

Benthic boundary layer flow beneath a large amplitude internal solitary wave

Benthic boundary layer flow induced by large amplitude internal solitary waves (ISWs) is investigated. Experimental measurements of the velocity fields close to the bottom boundary are presented to illustrate the generation of an unsteady boundary jet along the bed beneath an ISW of depression. The formation of the jet, the structural characteristics of which show striking similarities with those predicted by numerical model studies by Diamessis & Redekopp (2006), is attributed to boundary layer separation in the adverse pressure gradient region of the wave-induced flow. Moreover, experimental evidence is presented in support of the theoretical prediction of Diamessis & Redekopp (2006) for wave-induced vortex shedding at the lower solid boundary as a result of global instability. Measurements of the velocity field close to the bottom boundary illustrate coherent periodic shedding of vortex structures at the lower boundary in the adverse pressure gradient region aft of the wave. The vortical structures ascend high into the water column and cause significant benthic turbulence. It is shown that global instability has a critical threshold dependent upon the Reynolds number of the flow and the amplitude of the wave. The critical amplitudes observed are approximately half that predicted by Diamessis & Redekopp (2006) indicating internal wave-induced benthic mixing may be even more prominent than previously thought.

The benthic boundary layer flow induced by an ISW of elevation is less clearly understood. There is debate in the current literature as to whether ISWs of elevation can induce vortical structures and re suspend sedimentary material in a similar fashion to ISWs of depression (see Statsna & Lamb 2002 and Diamessis & Redekopp 2006). Recent experimental findings will be presented and discussed in the context of this debate.

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Melissa Coman
Australian National University

Horizontal convection forced by two spatially-separated regions of destabilising flux

We present horizontal convection experiments in which the thermal forcing is such that there is one region of stabilising buoyancy flux but two regions of destabilising buoyancy flux. We focus on the steady state circulation owing to the two plumes that form as a consequence of the destabilising fluxes. This arrangement of buoyancy fluxes is motivated by the high latitude sinking in the Northern and Southern Hemispheres of the current ocean. The experiments broadly outline the interaction of two deep sinking plumes and describes how the circulation and interior stratification changes when the ratio of the destabilising ‘surface’ buoyancy fluxes change. We classify the flow into three regimes of overturning, according to the pattern of interior circulation and depending on the relative strengths of the two plumes. We find that unequal plumes increase the interior stratification above that of two equal plumes, and when one plume is stronger than the other by more than 10%, the interior stratification is set by the stronger plume. We also introduce a sill into the horizontal convection experiment. The gap above the sill must accommodate a mass exchange in the thermal boundary layer and the return flow over the sill. We find that for a sill to have a significant effect on the circulation and heat transport the gap above the sill must be less than 25% of the depth of the domain.

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Georges Djoumna
University of Waterloo

An efficient semi-Lagrangian advection scheme for nonlinear shallow-water models on unstructured grids

The advection process strongly affects the evolution of temperature, salinity, and passive tracers in the ocean interior. It is related to the stability, accuracy, and efficiency of the numerical methods that are used in the ocean models. Here, the finite-element, semi-implicit, and semi-Lagrangian methods are used on unstructured meshes to solve the nonlinear shallow water system. A new semi-Lagrangian scheme is developed by tracking back a particle located at a quadrature node. It is shown that standard interpolation schemes such as linear and quadratic perform quite well with this alternative semi-Lagrangian method. A class of high order C1 interpolating schemes based on the Hsieh-Clough-Tocher finite-element is also developed for an accurate treatment of the advection terms. By tracking the characteristics backward from both the interpolation and quadrature nodes and using C1 interpolating schemes, an accurate treatment of the nonlinear advection terms and hence, of Rossby waves, is obtained. Numerical results of the test problems to simulate, the linear advection of a cosine hill and the slowly propagating Rossby modes are presented that demonstrate the performance of the proposed approach on more realistic problems.

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David Dritschel
University of St Andrews

Jet sharpening

Jets - narrow currents of streaming fluid - are found widely in planetary atmospheres and in the oceans. They are a product of fluid dynamical nonlinearity, whereby gradients of a nearly conservative dynamical tracer, potential vorticity, are steepened in places by Rossby wave breaking and homogenised in others by turbulent mixing. This talk illustrates, in a simple model, how jets intensify or sharpen as a result of basic barotropic and baroclinic instabilities.

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Jerome Fontane
University of St. Andrews

Two-dimensional forced turbulence

Two-dimensional turbulence has been extensively studied as it represents the simplest model for geophysical flows and it also exhibits one of the most remarkable features of fluid dynamics, namely the dual cascade of energy and enstrophy. When the flow is fed with an energy input at some given wavenumber kf , energy cascades upscale partly through the growing of vortices and enstrophy cascades downscale through vorticity filamentation. Since the pioneering theoretical work of Kraichnan (1967) and Batchelor (1969), it is well recognised that the energy spectrum takes the form E(k) = C.2/3k.5/3 for k < kf and E(k) = Cā2/3k.3 for k > kf where . and ā are respectively constant energy and enstrophy fluxes. However, Scott (2007) has recently pointed out the nonrobustness of this spectral form especially when the forcing wavenumber is well separated from the dissipation range and the Reynolds number is large. In that case, the energy-cascading range steepens from k.5/3 to k.2. High resolution numerical simulations using two different algorithms (spectral methods and contour advection) confirm the departure from Kraichnanfs spectral prediction of the k.5/3 energy-cascading range. The spectra are seen to converge very rapidly to a fixed form, then only evolving in time at large scales. We propose an analytical form for the enstrophy spectrum whose parameters are determined from the total energy and enstrophy evolution. The front of the enstrophy spectrum evolving towards low wavenumbers is consistent with the linear growth of energy. Finally, we discuss the spectrum shape by investigating the role played by filaments and coherent structures, the latest being responsible for the departure from Kraichnanfs theory.

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Stephen D Griffiths
University of Leeds

Modelling internal tides in the ocean

Internal tides are generated as the surface (barotropic) tide flows over topography. The implied vertical motion at the sea-floor can force large amplitude internal waves in the stratified ocean interior, with vertical displacements of 50m or more. At sufficiently low latitudes, this internal tide propagates away from the generation region, leading to an important energy transfer from the barotropic tide to the ocean interior.
Here we are interested in modelling both the generation of internal tides, and their feedback on the barotropic tide (an internal tide drag). To do this, a model is required for three-dimensional free-surface flow over quite general topography. We develop the linear hydrostatic framework of Griffiths and Grimshaw (JPO 2007), which uses a modal decomposition of the vertical structure of the flow fields. It is valid for arbitrary vertical stratification and large amplitude topography, in contrast to standard formulations which employ a linearised bottom boundary condition.
We show examples of how this formulation can be used to model (linear) tides, both locally and globally. Local solutions for internal tide generation over idealised topography (calculated with up to 256 vertical modes) can resolve fine-scale wave-beams emanating from steep topographic slopes. Global solutions for the internal tide (calculated with just a few vertical modes) are shown to account for much of the observationally constrained internal tide dissipation. Finally, global solutions of fully coupled barotropic and internal tides will be presented.

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S. G. Gopalakrishnan
Hurricane Research Division, AOML/NOAA, Miami, Florida

Recent Developments in Hurricane Structure and Intensity Forecasting Research at NOAA

Forecasting intensity changes in Tropical Cyclones (TCs) is a complex and challenging multi-scale problem because the factors that are known to influence these changes may vary in scales ranging between several hundreds of kilometers (example: environmental shear and upper ocean structure) to a few kilometers (example: vortex scale interactions and ocean waves) and, perhaps sometime, even down to a few hundred meters (example: individual cloud and turbulent scale motions). The hurricane forecast improvement project (HFIP) is a unified NOAA approach to guide and accelerate improvements in hurricane track, intensity and structure forecasts, with an emphasis on rapid intensity change. An integral component of the HFIP will be the development of improved coupled atmosphere-ocean-land surface, high-resolution non-hydrostatic regional models.A general introduction of the NOAAs high-resolution version of the hurricane prediction system, called the HWRFX, will be presented. The prediction system appears to produce some of the salient features observed in the inner core at about 3 km resolution. Within the context of this system some of the basic mechanisms leading to modeled intensity changes as a function of horizontal grid resolution will also be discussed. Our initial finding indicates that better resolution may be important to improve forecast of vortex scale motions but the hurricane intensity change forecasting problem is simply beyond a model grid resolution or a multi-processor problem. Better representation of physical processes in numerical models and improved initial conditions may lead to improved intensity forecasting.

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Greig Henderson
University of St Andrews

Jets in planetary-scale turbulent flows

This present work focuses on one key feature of large-scale flows in planetary atmospheres: high speed currents of fluid or 'jets'. We present the results from a series of freely-decaying quasigeostrophic turbulence simulations conducted to investigate the emergence of jets and their characteristic spacing. These numerical experiments have been carried out using a very accurate 'contour-advective semi-Lagrangian' (CASL) algorithm in order to simulate these complex flows, the evolution of which are controlled by two basic parameters, namely the Rossby deformation length and the planetary vorticity gradient. We discuss the problem of defining jets and determining their characteristic spacing and present evidence of persistent vortices co-existing with jets at long times, a result which few past studies have observed without forcing.

Coauthors: David G. Dritschel

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Paul Kushner
University of Toronto

Climatic influences on mixing in small lakes

This talk will present some ideas on the question of how climate variability and change influences the process of mixing in small lakes. In particular, we are interested in the thermal- and wind-forced vertical flux of heat and nutrients by eddying motions in small lakes that are capable of developing seasonal stable thermal stratification. Lake mixing depends nonlinearly on meteorological fields like the surface wind and temperature; because of this nonlinearity, the average meteorology does not necessarily reflect the average lake mixing. Because meteorological variability relevant to lake mixing often results from continental or even planetary scale atmospheric teleconnections like the PDO and AO/NAM, lake mixing is also a process whose control is highly nonlocal. We discuss these issues in the context of a recent study of summertime lake mixing dynamics at Toolik Lake in Northern Alaska. At this site summertime mixing is strongly controlled by the frequency and intensity of the rapid temperature drops and extreme wind events associated with passing cold fronts. Because of the nonlinear relationship between meteorological forcing and surface winds and temperatures, these subseasonal timescale (less than two week timescale) events are able to dominate the lake mixing, more than variability of the mean climate. Control of subseasonal variability is, in turn, linked to large-scale patterns of circulation. We will discuss how the lessons learned in this case study might apply to other studies of mixing in small lakes, and hope to generate a discussion on how research in this area might be further pursued collaboratively between climate scientists and limnologists.

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Kevin Lamb
University of Waterloo

Energetics of Solitary-like Internal Waves in the Ocean

Large amplitude, internal solitary-like (ISWs) waves are highly energetic phenomena common to the coastal ocean. They can contain a
significant amount of energy and play an important role in many processes including mixing, particle transport, and sediment resuspension. ISWs are often of sufficient amplitude that a correct evaluation of the energy flux associated with them requires the inclusion of fluxes of kinetic and available potential energy. In this talk I will discuss the use of an available potential energy density to calculate energy fluxes associated with large amplitude, highly-nonlinear internal waves. Applications to shoaling ISWs and to the generation of ISWs by tide-topography interaction will be presented.

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Dan Lucas
University of St Andrews

A family of helically symmetric vortex equilibria.

Vortices are ubiquitous phenomena in the atmosphere and oceans and our understanding of geophysical flows relies on a thorough knowledge of the dynamics of such structures. To such an end vortex equilibria have proved an invaluable tool in helping to develop fast and accurate mathematical models of a wide variety of fluid regimes. In this work we present a new class of vortex equilibria possessing helical symmetry. Such helical vortices are observed in a number of environmental applications, most notably in rotor wakes (e.g. wind turbines, propellors). Coherent columnar vortices (e.g. tornados) can also be observed to evolve with a twisting structure.

We define our steadily rotating equilibrium states by contours bounding a region of uniform axial vorticity in an incompressible inviscid irrotational unbounded fluid. We consider single and multiple vortex equilibria, parameterised by mean radius and centroid position, and examine stability properties using a helical CASL algorithm.

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Rebekah Martin
University of Manitoba

Analysis of a Major Arctic Storm Event During the STAR Campaign

The Storm Studies in the Arctic (STAR) is a four-year research Network (2007-2010) involving a wide range of activities on the part of researchers from five Canadian universities and Environment Canada. The project is concerned with the documentation, better understanding and improved prediction of meteorological and related hazards in the Canadian Arctic. As part of the project, a major meteorological field campaign took place from October 10 –November 30, 2007 and in February 2008 and was focused on southern Baffin Island, Nunavut, Canada. During the fall study period of the campaign, a major storm event occurred over the southern Baffin Island region from 16-19 November, 2007. During the system’s passage, over the Hudson Strait, several drop sondes and microwave measurements of its warm front were taken. This talk will provide an overview of the structure of this Arctic system as revealed by these measurements. As well, we will discuss a comparison of the archived forecast model output to the measurements.

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David J Muraki
Simon Fraser University

A Potential Vorticity Dynamics for Shallow Water on the Sphere

The evolution of weather systems in the midlatitude atmosphere are well-explained by the theory of quasigeostrophy (QG), in which slow, synoptic-scale airflows are described through the advection of potential vorticity (PV). The mathematics of QG is often justified by a limit of small Rossby number. However, this assumed limit is made invalid across the equator by the vanishing of the Coriolis effects.

A model based upon the dynamics of PV is developed for rotating shallow water on the sphere. Specifically, a PV-streamfunction relationship is defined which determines the flow velocities for the entire sphere. At midlatitudes, the fluid dynamics are equivalent to the beta-plane theory of QG, in the usual small Rossby number sense. In the equatorial regions, wave propagation at short-scales mimics the dispersion relation for equatorial beta waves. These dynamics compare favorably with computations of the equatorial crossing of topographic waves by Grose & Hoskins (1979).

Despite that this PV model is not obtained in the usual manner of small Rossby number asymptotic analysis, the propagation of mesoscale waves across the equatorial region retains QG-like accuracy. The PV dynamics are contrasted with the shallow water primitive equations from the perspectives of ray theory and baroclinic instabilities.

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Julie Pietrzak
Delft University

Stratified flows and their role in controlling the exchange between rivers and ocean basins

Estuaries, with their associated coastal river plumes, are the interface between fresh river waters and saline coastal waters. Much of the exchange between the two systems is controlled by stratified flows within the estuary and along the coastal zone. These Regions of Freshwater Influence (ROFI's) dominate the transport of freshwater and matter in coastal oceans. They control the exchange of freshwater, sediment, contaminants and nutrients between inland rivers and ocean basins. Consequently they have a huge impact on the health and biological productivity of coastal seas. They also control the ultimate fate of increased river runoff due to climate change, as it makes it way to the oceans. Yet they are poorly resolved in large scale models. We present a new upwelling mechanism; as detected in a unique sequence of sea surface temperature satellite images. These are the first images to show coastal upwelling induced by tidal straining. We also use a numerical model to show how this tidally driven upwelling mechanism comes into being. We discuss the implications of tidal straining induced upwelling on coastal dynamics and its impact on the fluxes of sediment and nutrients in the coastal zone. We also examine the role of wind driven upwelling and straining on stratification and mixing. Having shown the importance of this narrow coastal zone, we then explain why it should be resolved in ocean models and briefly introduce our unstructured grid model developments. We show that coastal and estuarine models based on unstructured meshes have distinct advantages over traditional Cartesian based models.

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Francis Poulin
University of Waterloo

The Instability of Time-Dependent Baroclinic Shear Flows

Geophysical shear flows in nature are often idealized to be steady. This approach has proven useful in predicting the stability characteristics of many flows such as the Gulf Stream, but it is not without limitations. As time variations are ubiquitous in nature, it is of great interest to understand how they can alter the stability of a system. Here, I present examples of aperiodic systems in order to better understand what effect irregularity (stochasticity) has on the stability characteristics of a basic state. In particular, I discuss the Mathieu's equation and the linearized Phillips model.

Both models show that the parametric mode, which is the unstable mode in the periodic limit, also exists in the presence of small to moderate stochasticities. The effect of the noise is to reduce the growth rate while extending the region of instability. Moreover, at moderate values of stochasticity and small mean frequencies there is a new mode of instability, the stochastic mode. This mode is completely absent in the periodic and white noise limits and achieves its largest growth rates for moderate degrees of stochasticity. This is another example of how irregularity can actually destablize a system. To compliment these findings, I present the results from numerical simulations that illustrate the fully nonlinear evolution of the parametric and stochastic modes.

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Marek Stastna
University of Waterloo

Two examples of multiscale simulations in stratified flow over topography

In this talk I will discuss the simulation of two different problems in stratified flow over topography that span widely disparate length scales. In the first problem, I consider the effects of rotation on supercritical flow over isolated topography. For appropriately tuned inflows, with rotation turned off, the simulations yield large disturbances that are trapped over the topography. Rotation allows a tail of long long, hydrostatic Poincare waves to form behind the initial disturbance. In certain cases a secondary non-hydrostatic disturbance can form well behind the topography (50 kilometers in some cases). In the second problem, I consider the effects of boundary layer separation on the generation of internal wave beams by subcritical flow such as a seiche over short topography. Throughout I will discuss the computational hurdles that needed to be overcome and will speculate on the sort of model improvements that would allow the greatest gains.

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G E Swaters
University of Alberta

Modal interpretation for the Ekman destabilization of inertially-stable baroclinic flow in the Phillips model

The role of dissipation in the transition to instability in baroclinic quasi-geostrophic flow can be counter-intuitive (Klein and Pedlosky, JPO, 1992). It is natural to assume that dissipation acts to reduce the growth rates of baroclinic flows that are inertially (i.e., in the absence of dissipation) unstable and for flows that are inertially stable, dissipation will lead to the decay in the perturbation amplitudes over time. However, it has been known since Holopainen (Tellus, 1961) and, within the context of the Phillips model, Romea (JAS, 1977) that sub-critical baroclinic shears in the linear inertial stability theory can be destabilized by the presence of an Ekman boundary layer and that this destabilization occurs even in the zero dissipation limit of the frictional theory. That is, there is a range of sub-critical baroclinic shears (in the linear inviscid theory) which are destabilized by the presence of an Ekman boundary layer no matter how small the Ekman number is. Recent work by Kretchetnikov and Marsden (Rev. Mod. Phys., 2007 and Arch. Rat. Mech. Anal., 2009) has described this counter-intuitive dissipative destabilization within the context of the underlying Hamiltonian structure of the (inviscid) model equations. In particular, Kretchetnikov and Marsden (2009) have extended Romea's (1977) work and showed that Ekman destabilization within the Phillips model can occur for baroclinic shears that are inertially nonlinearly stable in the sense of Liapunov.
It is important to appreciate that the discontinuous behavior of the zero dissipation limit of the marginal stability boundary when an Ekman layer is present cannot be dismissed as akin to the well known property that solutions to the Orr-Sommerfeld equation need not necessarily reduce to solutions to the Rayleigh stability equation in the infinite Reynolds number limit. The infinite Reynolds number limit of the Orr-Sommerfeld equation is singular in the sense that the order of the differential equation changes from fourth-order to second-order so that the mathematical properties of the allowed solutions cannot be expected to depend continuously as the Reynolds number increases without limit. This is not the case for the Phillips model with Ekman layers as the zero dissipation limit is not singular.
However, the "physical reason" for the Ekman-induced destabilization of inertially stable baroclinic quasi-geostrophic flow has yet to be given. The principal purpose of this talk is to provide the modal interpretation for the onset of this instability. By exploiting the concept of the "kinematic wave" introduced by Lighthill and Whitham (Proc. Roy. Soc. Lond. A, 1955), it is shown that the onset of Ekman destabilization of inertially stable baroclinic flow in a zonal channel in the Phillips model occurs when the dissipative kinematic wave phase velocity lies outside the range of zonal phase velocities spanned by the neutrally-stable planetary Rossby waves. The onset of dissipative destabilization does not correspond to a coalescence of the barotropic and baroclinic modes as in inviscid baroclinic instability and indeed the necessary conditions for baroclinic instability need not hold. This is the reason why sub-critical shears in the inviscid theory can be unstable even in the zero dissipation limit when Ekman layers are present.

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Chuong Tran
University of St Andrews

The number of degrees of freedom of three-dimensional Navier-Stokes turbulence

In Kolmogorov's phenomenological theory of turbulence, the energy inertial range scales with the wave number k as k-5/3 and extends up to a dissipation wave number kn, which is given in terms of the energy dissipation rate e and viscosity n by kn ? (e/n3)1/4. This result leads to Landau's heuristic estimate for the number of degrees of freedom that scales as R9/4, where R is the Reynolds number. Here we consider the possibility of establishing a quantitative basis for these results from first principles. In particular, we examine the extent to which they can be derived from the three-dimensional Navier-Stokes system, making use of Kolmogorov's hypothesis of finite and viscosity-independent energy dissipation only. It is found that the Taylor microscale wave number kT (a close cousin of kn) can be expressed in the form kT ? CU/n = (CU/\norm)1/2(e/n3)1/4. Here U and \norm are, respectively, a "microscale" velocity and the root mean square velocity and C ? 1 is a dynamical parameter. This result can be seen to be in line with Kolmogorov's prediction for kn. Furthermore, it is shown that the minimum number of greatest Lyapunov exponents whose sum becomes negative does not exceed R9/4, where R is defined in terms of an average energy dissipation rate, the system length scale, and n. This result is in a remarkable agreement with the Landau estimate, up to a presumably slight discrepancy between the conventional and the present energy dissipation rates used in the definition of R.

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Michael Waite
University of Waterloo

Buoyancy scale dynamics in stratified turbulence

Much of the research in stratified turbulence over the last two decades has been motivated by the kinetic energy spectrum of the atmospheric mesoscale, which is observed to have a -5/3 spectral slope. It has been shown recently that such a spectrum can result
from a downscale energy cascade in numerical simulations of stratified turbulence, provided that sufficiently high vertical resolution is employed. Specifically, the spectrum requires that vertical scales on the order of the buoyancy scale U/N be outside the dissipation range; here U is the root mean square velocity, and N is the buoyancy frequency. In practice, this constraint often necessitates the use of anisotropic numerical grids, in which vertical scales of O(U/N) are resolved but horizontal scales are not. In this talk, we will discuss
some of the features of stratified turbulence that develop on horizontal scales on the order of the buoyancy scale, and the consequences of neglecting them. The breakdown of coherent vortex dipoles undergoing the zigzag instability, as well as fully-developed turbulence forced by large-scale vortical modes, will be considered.

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Matthew Wells
University of Toronto

A theory to explain the entrainment ratio of gravity currents

The dynamics of both density and turbidity currents are determined by mixing and drag due to turbulence at the upper and lower interface. The magnitude the mixing and interfacial drag is often parameterized in terms of a dimensionless entrainment ratio E in gravity currents, but there has not previously been any unifying theory to predict either the magnitude of E or to determine how E depends upon the stability of the flow. Such a theory would then predict when the bottom drag co-efficient CD would be greater of less than the interfacial drag E, for different Reynolds and Froude numbers. We present an explanation of the functional dependence of E upon the Froude number and Reynolds number of a gravity current. Our theory is based upon the observed variation of the flux coefficient G (sometimes known as the mixing efficiency). Our main theoretical result is that E= 0.25 G Fr2 cos(?), where ? is the angle of the slope over which the gravity current flows, and Fr the Froude number. In the case of high Froude numbers we find that E ~ 0.1, consistent with observations of a constant entrainment ratio in unstratified jets and weakly stratified plumes. For Froude numbers close to one, G is constant and has a value in the range of 0.1 - 0.3, which means that E ~ Fr2, again in agreement with observations, and previous experiments. For Froude numbers less than one, G decreases rapidly with Froude number, explaining the sudden decrease in entrainment ratios that has been observed in all field and experimental observations. We also show that the functional form of the stratified turbulent diffusivity K? has the same dependence upon the Froude number as the entrainment ratio.

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Ram Yerubandi
National Water Research Institute, Environment Canada, CCIW

Some recent hydrodynamic modelling activities of the Great Lakes at Environment Canada

The Laurentian Great Lakes have horizontal scales of hundreds of kilometers and depth scales of hundreds of meters. These lakes are subjected to many of the same forcing as coastal oceans and serve as model basins for understanding the complex coastal ocean dynamics. An understanding of the hydrodynamic processes is essential to develop science based integrated management of the Great Lakes. With this broad objective in mind systematic monitoring and modeling studies of the North American Great Lakes have been carried out for well over three decades at Environment Canada's National Water Research Institute. In the first part of the talk I will briefly describe some historical modeling activities in the Great Lakes.
In the second part of the talk recent activities of coupling of weather prediction models with lake models will be discussed. Because of the size of the lakes, the Great Lakes have a profound influence on the local and regional weather and climate. Currently, the lake-ice component of the operational atmospheric system in the Canadian Meteorological Centre is treated as static, with the water surface temperature values being specified based on observations for determining heat fluxes into the atmosphere at the surface of the lake. However the air-lake interface is dynamic with momentum exchange, heat exchange and moisture/water exchange. As part of the project several hydrodynamic models are being tested on the lower Great Lakes. An intercomparison of these models with a time series observations of circulation and temperature in Lake Ontario showed promising results. Efforts are underway to dynamically couple one of these models with the operational weather forecasting system.

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Contributed Talks Abstracts

Payam Aghsaee
Queens University

Breaking of Shoaling Internal Solitary Waves

The breaking of fully nonlinear internal solitary waves shoaling upon a uniformly sloping boundary was investigated using two-dimensional direct numerical simulations. Our simulations were limited to narrow-crested waves, which are shown to be more common in geophysical flows. The simulations were performed for a wide range of boundary and wave slopes (0.01 < S < 0.3) extending the parameter range considered in previous laboratory and numerical studies. Over steep slopes (S > 0.1), three distinct breaking processes were observed; surging, plunging and collapsing breakers which are associated with reflection, convective instability and boundary layer separation, respectively. Over mild slopes S < 0.05 nonlinearity varies gradually and fission results from dispersion. The dynamics of each breaker type were investigated and the predominance of a particular mechanism was associated with a relatively rapid developmental timescale. The breaking location was modelled as a function of wave amplitude (a), characteristic wave length (Lw) and the isopycnal length along the slope (Li). The breaker type was characterized in wave slope (a/Lw) versus S space and the reflection coefficient (R), modelled as a function of the internal Iribarren number, was in agreement with other studies. The effects of grid resolution and Reynolds number R on (R), boundary layer separation and the evolution of global instability were considered. High Reynolds numbers (R > 104) were found to trigger a global instability, which modifies the breaking process, relative to the lower Re case, but not necessarily the breaking location and results in an increase in the reflection coefficient by approximately 10%.

Co-authors: Leon Boegman (Queens University), Kevin Lamb (University of Waterloo)

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Abbas Dorostkar
Queen's University

Three Dimensional Modeling of Internal Waves in a Medium Sized Lake

The three-dimensional (3D) high-resolution simulation of lakes with diameter > 1 km has not been performed, with sufficient resolution to resolve small-scale nonhydrostatic processes (e.g. high-frequency nonlinear internal waves), due to the computational requirements and challenges associated with parallel computing. In this research, the parallel 3D MITgcm is applied to simulate the multi-scale nonhydrostatic response of Cayuga Lake (a medium sized lake in central New York State) to the applied surface wind forcing. The model is validated on a 400 m x 400 m grid against field observations over an 11 day simulation, using various equations of state, drag coefficients and sub-grid scale closure schemes. The phase and amplitude of the basin-scale seiche dynamics were well modeled and the r.m.s. error between model results and simulations was minimized with a Smagorinsky/Leith scheme for the horizontal eddy viscosity and a constant vertical eddy viscosity of 10-3 m/s2 with a high order nonlinear equation of state. Using the parameters obtained from the basin-scale calibration, higher resolution hydrostatic and nonhydrostatic simulations were performed on a fine 39x39 m grid. Preliminary nonhydrostatic simulations resolve progressive nonlinear internal waves in the 10-4 Hz frequency band, but were unable to simulate shear instabilities, which have ~ 20 m wavelengths, well below model resolution. The MITgcm thus reproduces the spectrum of internal waves in Cayuga Lake ranging from the basin-scale to the nonhydrostatic features.

Co-authors: Andrew Pollard (Queen's University), Peter Diamessis (Cornell University)

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Michael Dunphy
University of Waterloo

The Influence of Mesoscale Eddies on the Internal Tide

The barotropic tide dissipates a well established estimate of 2.5 TW of energy at the M2 frequency. Bottom topography is responsible for a lot of this dissipation, and the generation of the internal tide is also a substantial sink of this energy. The fate of this energy is largely described by a cascade from large scales to smaller scales by non-linear wave-wave interactions until the energy is dissipated at turbulent length scales.
This thesis aims to investigate how the presence of mesoscale eddies (vortices) in the ocean affect the internal tide. The MITgcm is used to simulate internal wave generation by barotropic flow over topography. Baroclinic eddies are analytically prescribed and then geostrophically adjusted also using the MITgcm. Finally, the two are combined, and the internal tide field is analysed with and without the presence of eddies of various magnitude and length scales.
The results of this investigation does not find a strong transfer of energy between modes; the modal distribution of energy in the internal tide remains the same when an eddy is added. However, focusing and shadow beams of internal waves are produced in the wake of an eddy as the internal waves pass through it. The beams show very strong variations in intensity, vertically integrated energy flux can reduce almost to zero in the shadow regions and increase more than double in the focusing regions.
Modal decomposition of the horizontal flow field reveals that mode 2 and 3 waves are most strongly affected by the eddies and contribute strongly to the formation of the beams. Mode 1 appears to be less affected by the eddy. The larger wavelength and faster group velocity of mode 1 supports the notion that the eddy interacts with it less.

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V. Gerasik
University of Waterloo

Complex group velocity in absorbing media

Complex group velocity is common in absorbing and active media. From the physical point of view the concept of the complex group velocity is obscure. Unlike purely real group velocities in conservative dynamical systems, complex group velocity cannot
be associated with the velocity of energy transport, and the definition of the complex group velocity from energy principles is not established. The presented analysis expounds the connection between the complex group velocity and energy transport characteristics
for a class of hyperbolic, dissipative systems. The approach presented stems from the Lagrangian formulation, and is further illustrated with identities established for the damped Klein-Gordon equation; Maxwell’s equations, governing electromagnetic waves in partially conducting media; and Biot’s theory, governing acoustical wave propagation in porous solids.

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D. Godlovitch
University of Victoria

Monte Carlo Modelling of the Evolution of the Sea Ice Thickness Distribution

A Markov Chain Monte Carlo (MCMC) method for simulating the dynamic evolution of the thickness distribution of sea ice is introduced. It has been observed that the sea ice thickness distribution has a relatively invariant negative exponential form over data from a wide range of geographic locations and as yet, no model directly examines the physical mechanisms behind this property. The thickness distribution of sea ice results from a combination of thermodynamic growth and ice-ice interactions caused by forcing from winds and currents. The ice-ice interactions are complex and difficult to describe due to the material properties of sea ice. By simplifying the dynamics, a MCMC model is developed in order to explore the relative importance of the physical processes contributing to the observed thickness distribution.

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Wentao Liu
University of Waterloo

Lake Erie Modeling with ELCOM

Hydrodynamic and thermodynamic modeling of Lake Erie from April to October 2002 with ELCOM (Estuary, Lake and Coastal Ocean Model) is discussed in this talk. Observation data of inflows, outflows, winds, solar radiation, air temperature, and humidity are input hourly to the model. Three sizes of equally-spaced horizontal grids are implemented in the model, which range from 2 km, 1 km, to 600m. By comparing different grids, the preliminary results of the temperature and the velocities in the surface and vertical curtains crossing each basin are presented. Given the continuous mooring observation data for the temperature in one station in the eastern basin from May to October 2002, the simulation matches the observation quite well. There are also data available in several sampling stations in each basin, which contains the temperature profiles for four different days. A comparison of the observation and simulation using different sizes of grids is investigated as well.

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K. Mitchell
Simon Fraser University

Fourier Spectral Computing on the Sphere

Simulations of macro-scale geophysical fluid dynamics require an efficient method for computing on the surface of a sphere. I will present such a method that retains the essential simplicity of FFT-based methods in the 2D periodic domain. This is achieved by the natural periodic extension of the latitude coordinate, which maps the surface to a doubly periodic torus. In the Fourier basis for this mapping, many spherical differential operators are represented by sparse matrices lending to fast O(N2log(N)) implicit-explicit time-stepping for PDEs. The utility of this approach is demonstrated by applications to both diffusion- and wave-like PDEs. The MATLAB code for these demonstrations is available at http://www.math.sfu.ca/ kevmitch/pdeS.zip

Co-authors: David Muraki (Simon Fraser University), Andrea Blazenko

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M. Nica
University of Waterloo

Faraday waves in the Shallow Water Model

Faraday waves appear on the free surface of a fluid that is subject to sinusoidal vertical forcing. In this presentation, we study the generation of these waves in the context of the rotating reduced gravity shallow water model. By solving the linear stability problem we compute the regime of unstable wave numbers as well as their corresponding growth rates. Furthermore, we study the nonlinear evolution of these unstable modes to determine the effects of nonlinear equilibration. The effects of viscosity, Coriollis force, and stochastic forcing will all be considered.

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P. Pernica

Does the variability in wind driven turbulence drive variation in plankton patchiness?

Previous observations of the patchiness of zooplankton in the South Arm basin of Lake Opeongo (Ontario, Canada) have been linked to wind speed, with heterogeneous distributions occurring at wind speeds of 3 m/s - 7 m/s (Blukacz et al. 2009). The variations in wind velocity are thought to directly affect the structure of the mixed layer of the lake. In July 2008, velocity profiles of the water column of the epilimnion were measured over a one week duration. The effect of the wind on the mixed layer of the lake is quantified using the dimensionless Froude number (Fr) and the turbulent kinetic energy (TKE). Fr compares inertial forces to the strength of the stratification of the surface layer. Results indicate that the wind speed correlates with Fr. Wind speeds of 3 m/s to 7 m/s for which zooplankton patchiness was observed (Blukacz et al. 2009) correspond to Fr~1 indicating that basin is in a weakly mixed state. Average TKE in the epilimnion is shown to correlate with Fr and also displays spatial heterogeneity corresponding with Bluckacz (2007) observed change points of zooplankton variance.

Co-author: M.G.Wells

Blukacz, E. (2007). The Effect of Prey Patchiness on Zooplankton Growth Potential. PhD
thesis.

Blukacz, E. A., Shuter, B. J., and Sprules, W. G. (2009). Towards understanding the
relationship between wind conditions and plankton patchiness. Limnology and Oceanography,
54(5).

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Lynn Pogson
McGill University

Modeling ice algae in the Canadian Arctic Archipelago

Part of the sea ice ecosystem, ice algae are an important component of the carbon cycle in the Arctic, and can therefore have an impact on climate. We investigate the dynamics of an ice algae bloom by coupling an algae-nutrient model (Lavoie et al, 2005) with a thermodynamic sea ice model (Huwald et al, 2005). The sea ice component is a more sophisticated model than what has been used in past Arctic ice algae model studies, and it has a greater capacity to simulate ice algae communities present in different regions of the ice. To validate the model, we simulate an algal bloom at the base of the ice over a season and compare with data from the Resolute area in the Canadian Arctic Archipelago. Results are consistent with previous studies modeling this area. As well, an advection term in this model used to take into account movement between ice layers naturally handles the expulsion of algae during ice melt, and the loss in biomass triggered by melt at the base is more accurately simulated with this model than in past studies.

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T. Rees
University of Waterloo

Simulating Forced Waves In Continuously Stratified Fluids

In this talk I will discuss some of the problems that arise in numerical simulations where forcing the momentum equations is used to generate internal waves. In particular, I will show that when forcing waves in a fluid of nonconstant buoyancy frequency the expected waves in the resulting flow do not necessarily possess the most energy. I will explain this result using Sturm-Liouville theory, and present a method that forces the intended waves resonantly while avoiding resonance in undesired waves. This will be demonstrated in the context of a fully nonlinear pseudo-spectral model but the analysis naturally extends to any numerical method.

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Nancy Soontiens
University of Waterloo

Numerical Simulation of Trapped Internal Waves over Topography

Results from numerical experiments of a stratified Boussinesq fluid with background current are presented. The iterative numerical model includes a sheared background current and valley-like topography. Trapped waves of different amplitudes are produced for different types of background currents. A hysteresis loop is observed by changing the shear of the background currrent: significant differences in the amplitude of the wave develop if the shear is increased as opposed to decreased. Extensions to stratified non-Boussinesq fluids have been made.

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Derek Steinmoeller

University of Waterloo

A high-order numerical study of western boundary current separation along a curved coastline

Western boundary currents, such as the Gulf Stream and the Kuroshio, are of great interest because they contribute significantly to the pole-ward transportation of heat, chemistry and biology in the world's oceans. The dynamics of these currents are highly nonlinear and thus numerical simulations are perhaps the best tool available to study them. Recent advances in understanding the separation process of western boundary currents are due to simulations that integrated the barotropic Quasi-Geostrophic model using low-order numerical methods such as finite differences or the finite element method as in [1].
The goal of our study is to use a high-order spectral method to study the evolution of western boundary currents since they can better resolve the Munk layer that arises along the coastline. This is important because the boundary-layer dynamics are fundamental to the flow separation phenomenon. Results will be given for coastlines of various curvatures and for a wide range of Reynolds number and non-dimensional beta parameters. The previous aforementioned study focused on the separation of the time-averaged flow. This study is complementary in that we instead analyze the transient separation that occurs for relatively short times.

Co-authors: Francis Poulin (University of Waterloo), Serge d'Alessio (University of Waterloo)

1. Munday, D. R., and D. P. Marshall, 2005: Separation of western boundary currents from a cape. J. Phys. Oceanogr., 35, 1726-1743.

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Christopher Subich
University of Waterloo

A Pseudospectral Method for the Direct Simulation of the Incompressible Navier-Stokes Equations

Direct simulation of fluid flow in three dimensions is possible with modern computing resources. Demands on system memory scale as O(N^3), and this can be easily prohibitive. A pseudospectral method for these simulations has optimal memory characteristics, and even with a mixed Forurier/Chebyshev polynomial expansion allows single-timestep solution times with O(N^3 log(N)) complexity. This talk introduces such a model and presents preliminary results on the three-dimensionalization of a dipole/no-slip wall interaction.

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Paul Ullrich
University of Michigan

Riemann-Solver Based Shallow-Water FV Models on the Sphere

In this work we present a set of well-balanced second-, third- and fourth-order finite volume (FV) methods for solving the shallow water equations on the sphere using the MUSCL scheme of van Leer (1979), where edge fluxes are obtained via solving Riemann problems at Gaussian quadrature points along each edge. We compare three types of Riemann solver, including the basic solver of Rusanov, the scheme of Roe (1981) and the new AUSM+-up solver of Liou (2006). We also make use of the cubed sphere grid of Ronchi et al. (1996), which provides almost uniform resolution in all areas of the sphere using six identical grid patches obtained from "inflating" a cube that has been placed inside of a sphere. The resulting numerical methods are compared using several standard shallow water test cases, including the test suite of Williamson et al. (1992) and the unstable barotropic instability of Galewsky et al. (2004). As expected, we observe high-order accuracy from each of the finite volume schemes.


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