7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011 1

Three-dimensional simulation of NLIW generation, propagation and breaking in Cayuga Lake

Leon Boegman and Abbas Dorostkar

Environmental Fluid Dynamics Laboratory, Department of Civil Engineering

Queen’s University, Kingston, Canada K7L 3N6 Leon.Boegman@civil.queensu.ca

Abstract

Nonlinear internal waves (NLIWs) are ubiquitous features of coastal regions in lakes and oceans. NLIWs are typically investigated using direct numerical simulations or laboratory experiments, at low Reynolds number, or through two-dimensional and/or under-resolved three-dimensional (3D) field-scale numerical simulations. Here, we apply the 3D non-hydrostatic MITgcm to simulate the dynamics of NLIWs in long narrow Cayuga Lake (NY). This allows for high-resolution simulation (20 m x 20 m x 0.5 m) in a closed domain at high Reynolds number (Re ~ 105). Using approximately 300 million grid points, the model accurately reproduces the basin-scale internal seiche response, progressive surges and NLIWs over an 11-day simulation. Although the lake is long and narrow, sidewall and bottom topographic features readily reflect NLIW packets, causing them to propagate in both the longitudinal and transverse coordinate directions. Chaotic wave profiles result and subsequently shoal along both the basin sidewalls and the longitudinal ends of the basin. Internal wave dynamics in long narrow systems (e.g. Finger Lakes, lochs, fjords, estuaries and straits) are therefore fundamentally three-dimensional. 1. Introduction

Nonlinear internal waves (NLIWs) are ubiquitous features of lakes and coastal oceans (Boegman et al 2003; Jackson 2007). Shoreward propagating NLIWs will shoal upon sloping topography generating localized patches of energetic turbulence, mixing, nutrient flux and sediment resuspension (e.g., Klymak and Moum 2003; Hosegood and van Haren 2004). Although these processes are often investigated using two-dimensional simulations (e.g., Bourgault et al 2007; Vlasenko and Hutter 2002) and quasi two-dimensional laboratory scale flume experiments (Aghsaee et al 2010; Boegman and Ivey 2009), it is becoming increasingly apparent that NLIW generation, shoaling and breaking are three-dimensional (3D) processes, locally influenced by topography (Vlasenko and Stashchuck 2007; Zhang et al., 2011). Present computational power is not sufficient to fully capture the 3D non-hydrostatic dynamics occurring from wave generation to shoaling in oceanic domains. The applied thin-grid aspect ratios of horizontal to vertical spacing (i.e. Δx Δz, respectively) required to capture stratification lead to second-order accurate simulations being dominated by numerical dispersion when the grid leptic ratio, Γ = Δx/h1 < O(1), where h1 is the pycnocline depth (Vitousek and Fringer 2011). This is significant because NLIWs evolve from the flow under a balance between physical dispersion and nonlinear steepening of basin-scale motions (Horn et al., 1999; Boegman et al., 2005) and excessive numerical dispersion will non-physically

7th Int. Symp. on Stratified Flows, Rome, Italy, August 22 - 26, 2011 2

alter the waveform. Recourse to low Reynolds number simulations and/or experiments, which resolve all relevant scales, with suitable leptic ratios in physically smaller domains, remain problematic as they do not accurately capture large Reynolds number dynamics (e.g. Aghsaee et al., 2010). Cayuga Lake, a long narrow Finger Lake in upper New York State, is 62 km long, 3 km wide and has a maximum depth of 132 m. The presence of high-frequency nonlinear internal waves in neighbouring Seneca Lake (Hunkins and Fliegel 1973) and other similar lakes worldwide (e.g. Farmer 1978; Thorpe et al., 1972), their generation from oscillatory basin-scale seiches and ultimate interaction with topography, make this lake an ideal natural laboratory to study the ability of numerical simulation models to reproduce these complex dynamics. Attainable spatial resolutions are greater than what are achievable in the ocean (e.g. Vlasenko and Stashchuck 2007; Zhang et al., 2011) and Reynolds numbers are greater than what are achievable in the laboratory (e.g. Aghsaee et al 2010; Boegman and Ivey 2009). In the present study we apply a parallel non-hydrostatic Reynolds-averaged Navier-Stokes equation (RANSE) model to simulate NLIW generation, propagation and shoaling in Cayuga Lake. The objectives of the present paper are to: (1) accurately simulate NLIW dynamics at large Reynolds number; (2) visualize the effects of localized topography on NLIW propagation and shoaling; and (3) generalize these results to guide future simulation and field-observation of NLIW shoaling processes. 2. Methods

2.1 Model description

The numerical model used for this study is the non-hydrostatic 3D z-coordinate MITgcm, solving the Boussinesq form of the RANSE for an incompressible fluid on a staggered Arakawa-C grid. The code and further details are available online. The model employs a Smagorinsky formulation (with a multiplier of 0.7) to parameterize the horizontal eddy viscosity and the horizontal diffusivity was set to 10-7 m2s-1. The vertical eddy viscosity and diffusivity were computed by the K profile parameterization (KPP) scheme using background vertical viscosity and diffusivity of 10-3 m2s-1 and 10-6 m2s-1, respectively. The density was computed by the polynomial equation of state. The quadratic bottom friction applied a drag coefficient of 0.002. The wind stress fluxes at the free surface were calculated using the bulk formulae; this study is primarily concerned with wind forced dynamics and so thermodynamic heat transfers were neglected over the 11-day simulation. The Cartesian grid is uniform in the horizontal, with results presented using 23 m, 40 m and 450 m grid spacing. The vertical domain is

Figure 1: Cayuga Lake bathymetry showing 30 m isobaths. Thermistors were deployed at

S1, S2, S3 and S4. Wind rose (m s-1) measured at meteorological station in inset.

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discretized into 98 non-uniformly spaced z-levels ranging from 0.5 m between 10 and 30 m depths (through the thermocline), 1 m toward the surface and from depths 30 to 50 m, and 2.95 m from 50 m depth to the bottom. In total, the 3D domain is discretized with a maximum of 226 million computational cells. The model was initialized from rest and integrated over a period of 11 days (from day 269 until day 280 in 2006). The model was run on 50 to 80 Quad-core 2.52 GHz Sparc64 VII processors on Sun SPARC Enterprise M9000 Servers with a timestep between 1 s and 5 s to satisfy the CFL condition. The overall wall-clock runtime for the 11 day simulation on the 40 m grid was 4 months on 50 processors. 2.2 Field data

Data used in this study was obtained from a field experiment conducted from mid-September to mid-November of 2006 in Cayuga Lake (E. Cowen and S. Schweitzer, Cornell University). Temperature timeseries data were recorded at four thermistor chain moorings (S1, S2, S3 and S4 in Figs. 1 and 2) using SeaBird SBE-39 thermistors with a 25 sec sample interval a resolution of 0.0001 °C and an accuracy of ± 0.002 °C. The thermistors were vertically spaced at depths ranging from 10 m to 80 m through the water column (Fig. 2). Wind speed and direction was measured at 10 min intervals, 6 m above the water surface using a shore-based Campbell Scientific meteorological station located on an exposed headland (Fig. 1). 3. Results 3.1 Comparison between model and field data

The results from the 450 m grid have been validated against temperature observations using both qualitative and quantitative metrics (Dorostkar et al., 2010a,b). The model achieved a correlation factor of 0.7 and a basin-wide root-mean-square error of 1.9 °C, which is comparable to 3D model applications in other lakes (e.g. Huang et al., 2010). The 450 m model accurately reproduces basin-scale processes; including the linear basin-scale internal seiche (80 h period) and nonlinear basin-scale progressive surge fronts (not shown), but high-frequency motions are aliased by the grid (Dorostkar et al., 2010a,b). The 23 m model qualitatively reproduces NLIW packets at S2 and S3 (Figs. 3 and 4). The observed NLIWs have amplitudes 5 m < a < 20 m and wavelengths 181 m < λ < 228 m, as computed using the phase speed and Korteweg-de Vries theory (Boegman et al., 2003). This results in ~10 grid cells resolving both a and λ (10 < a/Δz < 40; 8 < λ/Δx < 10). The observed waves are generated earlier, or propagate slightly faster than modeled; potentially due to the choice of drag coefficient. The observed waves have more high-frequency wave content and spectral energy plots (not shown) suggest that the observed high-frequency oscillations result from shear instabilities, which have wavelengths ~20 m (Boegman et al., 2003), below our model resolution (see Botelho and Imberger 2007). The grid leptic ratio 1 < Δx/h1 < 2 and we expect numerical, relative to physical, dispersion to be minimal (Vitousek and Fringer 2011). 3.2 Three-dimensional visualizations

We visualize the shoaling behavior of the NLIW packets at the southeastern end of the basin (1.5 km < x < 5.5 km, Figs. 1 and 2 near S1). A longitudinal curtain (Fig. 5a-e) shows the NLIW packets from Figs. 3 and 4 shoaling and breaking along the southern lake shore (Fig. 5e), generating a vertical mode-two wave that propagates northward (Dorostkar and Boegman 2011). Visualization of the 10°C isosurface (Fig. 5k) shows the NLIW packet to interact with

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a submerged ridge, generating a secondary NLIW packet that propagates eastward (Fig. 5l) and eventually shoals on the eastern shore (Fig. 5m-n). The incident and reflected transverse wave packets are clearly visualized in a transverse curtain (Fig. 5f-j). Similar shoaling behavior occurs when the longitudinal surge and wave packet propagates through a bathymetric sill and contraction (7 km < x < 11 km, Figs. 1 and 2 near S6). Transverse NLIWs are generated from both shoreline and bathymetric features (Fig. 6a-c,f-h,k-m and Fig. 6c-e,h-i,m-o, respectively) and they shoal along the lateral boundaries.

Figure 2: Cayuga Lake profile, showing the depth along the lake thalweg and the locations of thermistor moorings. Distance from southeastern end is the x-coordinate in Figs. 5 and 6. 4. Discussion and Conclusions We have numerically modelled the generation, propagation and shoaling of NLIWs in Cayuga Lake over 11 days. The NLIWs were resolved using an unprecedented 226 million grid points; comprising >10 grid points over the wavelength and the amplitude. Dispersive wave generation is physically captured with a leptic ratio Δx/h1 ~ 1. These metrics and the physical domain size are similar to Vlasenko and Stashchuck (2007), who applied the MITgcm to simulate the shoaling and breaking of initialized NLIWs in the Andaman Sea. We achieve less than half their leptic ratio and directly simulate NLIW generation. We employ 30 times the grid points of Zhang et al. (2011), who attained the highest‐resolution simulations of internal waves in the South China Sea; a domain ~6 times that used in this study. They achieved leptic ratios between 14 and 5 and under resolved the wave amplitude using <6 grid points. These studies provide benchmarks and metrics outlining present computational limits for the 3D simulation of NLIW generation, propagation and shoaling. We conclude that NLIW propagation in long narrow lakes is fundamentally 3D, with secondary transverse NLIWs being generated from topographic features as the primary surge and NLIW packet propagates along the longitudinal axis. This fundamental 3D character means that 3D simulation is required even in systems such as narrow lakes, lochs, estuaries, straits and fjords, where the bathymetry is quasi two-dimensional.

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Figure 3: Observed and modelled temperature contours at station S2 from a simulation with a 23 x 23

m horizontal grid.

Figure 4: Observed and modelled temperature contours at station S3 from a simulation with a 23 x 23

m horizontal grid.

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Figure 5: Simulated (40 m grid) NLIW shoaling along the southern bathymetric slope (1.5 km < x < 5.5 km, Figs. 1 and 2 near S1). Panels (a)-(e) and (f)-(j) show longitudinal and transverse curtains at 19.65 km west and 5.04 km north, respectively, of temperature contours. White contours denote 10

and 13 °C. Panels (k)-(o) show a plan view of the 10 °C isosurface with lighting to highlight wave amplitude. Blue lines show locations of temperature contour curtains. Shoreline and 10 m

bathymetric contours are indicated with black lines.

(a) (f) (k) (b) (g) (l) (c) (h) (m) (d) (i) (n) (e) (j) (o)

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Figure 6: Simulated (40 m grid) NLIW shoaling through the sill and bathymetric contraction (7 km < x < 11 km, Figs. 1 and 2 near S6). Panels (a)-(e) and (f)-(j) show transverse curtains at 10.29 km

and 8.04 km north, respectively, of temperature contours. White contours denote 10 and 13 °C. Panels (k)-(o) show a plan view of the 10 °C isosurface with lighting to highlight wave amplitude.

Blue lines show locations of temperature contour curtains. Shoreline and 10 m bathymetric contours are indicated with black lines.

(a) (f) (k) (b) (g) (l) (c) (h) (m) (d) (i) (n) (e) (j) (o)

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Acknowledgements We thank E. Cowen and S. Schweitzer for providing the field observations and P. Diamessis for his encouragement. Funding was from NSERC Discovery Grants to L.B. and Andrew Pollard. A.D. was also supported by Queen’s University and the Huntly Macdonald Sinclair Fellowship. Computing was provided by the HPCVL and a CFI-OIT grant to L.B.

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