Ocean Mixing and Stirring  
  Principal Investigator: Dr. Miles A. Sundermeyer
msundermeyer@umassd.edu • 508-999-8892
University of Massachusetts Dartmouth
School for Marine Science and Technology
 
 

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Teaching:

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Research:

* Lidar Studies of Small- Scale Lateral Dispersion - LatMix
* 3D Dye Mapping using Airborne Lidar - Florida 2004 Pilot
* Lab Studies of Stirring by Small-Scale Geostrophic Motions
* Numerical Simulations of Vortical Mode Stirring
* Coastal Mixing & Optics (CMO)
(... More links coming soon!)
* The North Atlantic Tracer Release Experiment (NATRE)
(Links coming soon!)

Additional Links
* M. Sundermeyer's CV (PDF)

 

© 2005 Miles A. Sundermeyer (msundermeyer@umassd.edu)
Note: Please do not use the data, text, or images contained on this site without prior permission.

Numerical Simulations of Vortical Mode Stirring
PIs: M. A. Sundermeyer, M.-P. Lelong, and J. R. Ledwell
Grant Title: Numerical Simulations of Episodic Mixing and Lateral Dispersion by Vortical Modes
Funding Agency: Office of Naval Research
Award: $251,805

Objectives

Numerical simulations are used to investigate lateral dispersion caused by small-scale geostophic motions generated by the relaxation of diapycnal mixing events. Our long-term goal is to better understand rates and mechanisms of lateral dispersion in the ocean. The specific goals of this study are to:

  • Provide quantitative predictions of vortical mode stirring
  • Compare numerical simulations with results from CMO dye release studies
  • Provide a basis for parameterizing vortical mode stirring in ocean models

The Numerical Model

 
(Left) Time series of PE and KE spin-up from rest to a statisitcally steady state. (Right) Kinetic energy spectrum of fully spun-up model run.

We used a 3-D pseudo-spectral model (Winters et al., 2004) to solve the Boussinesq equations and an advection/diffusion equation for a passive tracer.* The model was run on a triply periodic domain, with typically 128x128 gridpoints in the horizontal, and either 64 or 128 gridpoints in the vertical. Typical domain size was Lx = Ly = 5000 m, Lz = 12.5 m, after rescaling. To limit the computation time required to resolve both buoyancy and inertial time scales, the model was run at reduced N/f = 20, compared to a more realistic value of 200. Nondimensionalization of the momentum equations shows that the dynamics of the vortical mode field are invariant under this scaling, provided that the Burger, Rossby, and Ekman number are all held fixed. Other parameters in the model (e.g., stratification, and the size of mixed patches) were based on observed values from late summer over the New England shelf.

The model was spun up from a state of rest and uniform stratification by injecting potential energy in the form of randomly placed Gaussian-shaped stratification anomalies. The anomalies were periodically introduced into flow at random locations in the model, according to a pre-determined rate of PE input (i.e. buoyancy flux). Once the model flow had equilibrated to statistically stationary state where PE input was balanced by viscous dissipation, dye was injected at the center of the model domain and tracked as a means of diagnosing lateral and vertical dispersion. The diffusivities were then compared for different model runs under different forcing parameters and used to test the theoretical scaling of Sundermeyer et al. (2005),

Spin-up of a typical model run, along with model fields over the course of the run are shown in the figures above. Results from a typical simulation are shown in the video below.

(Double click on image to view animation - 3.1 Mbyte.)
Typical model run with randomly placed density anomalies throughout the model domain. Top panels are plan views of dye, potential density anomaly, and PV with velocity vectors overlain. Bottom panels are vertical cross sections of the same variables.

Results

A series of model runs was performed using various values of the relevant parameters either alone or in combination in order to test the parameter dependence of Kh. Effective horizontal diffusivities diagnosed from model tracer were consistently about 6-10 times larger than those predicted based on geostrophic/random walk scaling. After accounting for this scale factor, model results for a wide range of parameter values were found to be consistent with the predicted scaling. These results are as follows.

 
(Left) Rate of change of 2nd moment of dye in x-direction, and (right) effective horizontal diffusivity for a series of model runs for varying event recurrence frequency showing an increase in Kh for increasing event frequency.

Event Frequency: The frequency of events was varied by an order of magnitude compared to our base run. For small f, the resulting Kh in the model varied approximately linearly with f, consistent with the geostrophic scaling prediction. For large f, a higher order dependence was found, indicating a transition to a more energetic parameter regime. The latter was most notably characterized by the failure of KE in the model to achieve statistically stationarity, and an apparent cascade of energy to large scales.

 
(Left) Rate of change of 2nd moment of dye in x-direction, and (right) effective horizontal diffusivity for a series of model runs for varying background viscosity showing a decrease in Kh for increasing viscosity.

Background Viscosity: Model viscosity was also varied by an order of magnitude. For large nu2, Kh again varied linearly, consistent with the predicted geostrophic scaling. For small nu2 (high Kh), a higher order dependence was again found, accompanied by unbounded growth of KE, and an inverse energy cascade.

Coriolis Frequency and Horizontal Event Scale, f and L: A variety of combinations of f and L were examined (see table), and found to be consistent with the theoretical scaling. Most notably, when f and L were increased by a factor of 2, and nu2 and f were simultaneously decreased by a factor of 2, the result was nearly identical (relative to the scaling) to the base run, i.e., the two runs were dynamically similar.

Buoyancy Frequency and Vertical Event Scale, N and h: A variety of combinations of N and h were examined, and also found to be consistent with the theoretical scaling. Furthermore, when N was decreased by a factor of 2 and h and nu2 simultaneously increased by a factor of 2, the result was again dynamically similar to the base run.
Parameter Predicted Kh Model Kh
f x 2, L x 2 Kh x 2 Kh x 1.8
f x 2, L x 2
nu2 x 2
Kh x 4
Kh x 4.3
f x 2, L x 2
nu2 x 2, phi x 2
Kh x 2 Kh x 2
N x 2, h x 2 Kh x 4 unbounded KE
N x 2, h x 2
nu2 x 4
Kh = same Kh = same

 

Strongly Nonlinear/Turbulent Regime:

 
(Left) Time series of PE and KE for a strongly nonlinear model run showing a steady build-up of energy throughout the run. (Right) Kinetic energy spectrum of fully spun-up model run showing an increase in energy at large scales, and a k-5 spectral slope.

The unbounded growth of KE found at high f and low nu2 is similar to the inverse energy cascade found in 2-D turbulence with coherent structures (note the characteristic k-5 energy spectrum). We hypothesize that the transition to this turbulent regime should occur when the frequency of events is of order or greater than the (1 / viscous time scale), so that anomalies are likely to interact nonlinearly, i.e.,

In practice, nonlinear interactions can still occur at f TkB < 1, since the near proximity of anomalies is sufficient for interactions to occur. Our model results suggest an actual threshold of f TkB <= 0.01-0.1.

 
Predicted versus modeled diffusivities for all runs combined showing agreement with the theoretical scaling to within a constant scale factor. Dashed blue line is 1:1 curve, solid red line is after multiplying the predicted values by a factor of 7.

Conclusions

Numerical simulations of vortical mode stirring were generally consistent with the parameter dependence predicted by Sundermeyer et al. (2005). Effective horizontal diffusivities were consistently about 6-10 times larger than predicted from geostrophic / random walk scaling. An additional parameter, the ratio of the vertical diffusion time scale to the inertial period, was shown to be important. Our results suggest that vortical mode stirring may be as much as an order of magnitude greater than the lower-bound prediction of Sundermeyer et al. (2005). An additional parameter regime characterized by a nonlinear energy cascade to large scales may lead to even greater dispersion.

 

Publications

Sundermeyer, M. A., J. R. Ledwell, N. S. Oakey, and B. J. W. Greenan, Stirring by Small-Scale Vortices Caused by Patchy Mixing. J. Phys. Oceanogr., 35, 1245-1262, 2005.
Abstract   PDF  
Sundermeyer, M. A., and M.-P. Lelong, Numerical Simulations of Lateral Dispersion by the Relaxation of Diapycnal Mixing Events. J. Phys. Oceanogr., 35, (12), 2368-2386, 2005.
Abstract   PDF  
Lelong, M.-P. and Sundermeyer, M. A., Geostrophic Adjustment of an Isolated Diapycnal Mixing Event and its Implications for Small-Scale Lateral Dispersion. J. Phys. Oceanogr., 35, (12), 2352-2367, 2005.
Abstract   PDF  
 

Prof. Miles A. Sundermeyer
The School for Marine Science and Technology
706 South Rodney French Blvd., New Bedford, MA 02744-1221
voice: 508.999.8892 fax: 508-910-6371 e-mail: msundermeyer@umassd.edu
www.smast.umassd.edu/msundermeyer