1. Linear and non-linear properties
of mesoscale eddies
Jim Price, WHOI
June, 2010
revised November, 2018
Mesoscale eddies observed in the subtropical gyres appear to propagate west in common with linear, long baroclinic Rossby waves: at 30 N, about 3 km/day. Eddies have a significant amplitude, thermocline displacements of about +-50 m, that implies
geostrophic currents considerably greater than this propagation speed. Do eddies
have significant nonlinear properties on account of their largish amplitude? To explore this we will solve a shallow water beta-plane model that is configured with an isolated mesoscale eddy that is seeded with floats and a passive tracer. Does the movement of the floats and the tracer accord with the movement of the eddy thickness anomaly?

2. Slide Index:
3) model setup
4) Case I: animation of finite amplitude anti-cyclone case
5) currents and tracer at 365 days
6) PV balance and eddy displacement
7) Case II: animation of small amplitude anti-cyclone case
8) currents and tracer at 365 days
9) PV balance and eddy displacement
10) Case III: currents and tracer at 365 days for a finite amplitude cyclone
11) PV balance and eddy displacement
12) Experiments with a wide range of eddy amplitude
13) Summary and remarks

3. In the three shallow water model numerical experiments shown here:
Ho = 500 m, density = 2 kg/m^3, eddy radius = 100 km.
Rd ~ 40 km, latitude = 30 N, beta plane.
2) Very small diffusion, and dx = dy = 10 km.
3) IC is a cylinder of thickness anomaly
h = 50 m, Case I, finite amp. anti-cyclone,
1 m, Case II, small amp. ant-cyclone, or
-50 m, Case III, finite amp. cyclone.
4) The eddy is released from rest. A
passive tracer has exactly the shape of the
initial thickness anomaly.
5) The scaling takes account of the sign
of the initial thickness anomaly and so h
always appears to be positive in the plots. IC

4. Case I: h = 50 m positive, finite amplitude anti-cyclone
Uscale = 0.3 m/sec
Field of velocity vectors, scaled with the speed = 0.3 m/sec shown at lower left. Color contours are thickness (pressure) and the moving dots are floats. Note that the time interval between frames is increased after time = 5 days.

5. Case I: h = 50 m positive, finite amplitude anti-cyclone
Uscale = 0.3 m/sec
Field of velocity vectors at 365 days, scaled
with the speed at lower left. Contours are
thickness (pressure) and moving dots are floats.
The passive tracer at 365 days, advected
and diffused during the integration.
After adjusting to near-geostrophy, the eddy begins translating (propagating)
westward, mainly. This fairly strong eddy carries along floats and passive tracer.

6. Case I: h = 50 m positive, finite amplitude anti-cyclone
Uscale = 0.3 m/sec
The balance of potential vorticity
along the line y = 0 at 360 days.
The displacement of the eddy peak in
east and north directions.
The PV balance is almost linear, and the eddy peak propagates just slightly
equatorward of due west. The westward speed is very close to the long Ro wave speed. The ratio stretching/relative vorticity and the evident east-west scale is also consistent with linear Ro wave dispersion.

7. Case II: h = 1 m positive, small amplitude anti-cyclone
Uscale = m/sec
Field of velocity vectors, scaled with the speed = m/sec shown at lower left. Color contours are thickness (pressure) and the nearly stationary blue dots are floats. Note that the time interval between frames is increased after time = 5 days.

8. Case II: h = 1 m positive, small amplitude anti-cyclone
Uscale = m/sec
In this small amplitude case, the overall evolution of the thickness and velocity are very similar to the previous case (amplitude aside). This eddy does not transport floats or passive tracer; it acts very much like a linear wave.

9. Case II: h = 1 m positive, small amplitude anti-cyclone
Uscale = m/sec
The balance of PV is very similar to that of the finite amplitude case. The
translation of this small amplitude eddy is due west.

10. Case III: h = -50 m negative, finite amplitude cyclone
Uscale = 0.3 m/sec

11. Case III: h = -50 m negative, finite amplitude cyclone
Uscale = 0.3 m/sec
The balance of PV is beginning to look familiar; the translation of this finite amplitude
cyclone includes a modest poleward component.

12. Eddy and float zonal speed
The results from nine experiments with anti-cyclones of varying initial amplitudes; Case I is ho/H = The zonal (westward) motion of the eddy peak (green line) is somewhat sensitive to eddy amplitude, being greater with greater amplitude. The ensemble average westward motion of floats started within the eddy (red line) is very sensitive to eddy amplitude, and is clearly a finite amplitude effect.

13. Summary and remarks:
Westward translation is a property of (i.e., is a consequence of, can be understood
from) PV balance. The PV balance is mainly linear, and is strongly scale dependent,
and the eddy disperses in space just like a Rossby wave packet; the peak translates westward
at very nearly the long Rossby wave speed, bRd2.
There is a small but noticeable poleward/equatorward translation of the eddy peak in the
large amplitude cases I and III, which is a finite amplitude effect. There is a comparable finite amplitude effect upon westward propagation, large amplitude eddies propagating slightly faster.
The transport of a passive tracer (floats) is entirely a finite amplitude effect, and appears to
scale reasonably well with ho/H.
Are these things waves or eddies? They are evidently both at once, depending upon the
amplitude and the property of interest. Mesoscale eddies share the nearly linear PV
balance and propagation of a Rossby wave packet, and yet are almost circular. Finite amplitude
eddies (realistic amplitudes) will thus readily trap and transport material.
In these initial value problems for an isolated eddy, the tracer transport appears to be
an incidental, or kinematic effect rather than a fundamental aspect of the eddy dynamics. That
might not hold over a long term when the eddies are embedded within and would presumably interact to some degree with a gyre-scale circulation.