



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 smallscale geostophic motions generated by the
relaxation of diapycnal mixing events. Our longterm 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 spinup from rest to
a statisitcally steady state. (Right) Kinetic energy spectrum of
fully spunup model run.

We used a 3D pseudospectral 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 Gaussianshaped stratification anomalies.
The anomalies were periodically introduced into flow at random locations
in the model, according to a predetermined 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),
Spinup 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 K_{h}. Effective horizontal diffusivities
diagnosed from model tracer were consistently about 610 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 xdirection,
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 xdirection,
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
nu^{2}, Kh again varied linearly, consistent with
the predicted geostrophic scaling. For small nu^{2} (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 nu^{2} 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 nu^{2}
simultaneously increased by a factor of 2, the result was again
dynamically similar to the base run.

Parameter 
Predicted K_{h} 
Model K_{h} 
f x 2,
L x 2

K_{h} x 2

K_{h} x 1.8

f x 2,
L x 2
nu^{2} x 2

K_{h} x 4

K_{h} x 4.3

f x 2,
L x 2
nu^{2} x 2,
phi x 2

K_{h} x 2

K_{h} x 2




N x 2,
h x 2

K_{h} x 4

unbounded KE

N x 2,
h x 2
nu^{2} x 4

K_{h} = same 
K_{h} = same 


Strongly Nonlinear/Turbulent Regime:




(Left) Time series of PE and KE for a strongly nonlinear
model run showing a steady buildup of energy throughout the run.
(Right) Kinetic energy spectrum of fully spunup 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 nu^{2}
is similar to the inverse energy cascade found in 2D 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 T_{kB} < 1, since the near
proximity of anomalies is sufficient for interactions
to occur. Our model results suggest an actual threshold of f
T_{kB} <= 0.010.1.




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
610 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
lowerbound 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 SmallScale Vortices Caused
by Patchy Mixing.
J. Phys. Oceanogr., 35, 12451262, 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), 23682386, 2005.

Abstract
PDF

Lelong, M.P. and Sundermeyer, M. A.,
Geostrophic Adjustment of an Isolated Diapycnal Mixing
Event and its Implications for SmallScale Lateral Dispersion.
J. Phys. Oceanogr., 35, (12), 23522367, 2005.

Abstract
PDF



