Here is described post-convective eddy restratification simulations using eddy density structures based on those in the 0.1o North Atlantic simulation near 50.87oN 27.75oW (the center of the southern lobe of maximum negative eddy vertical advection in the simulations).
The 111x111 Cartesian grid is centered at 50.87oN 27.75oW and has 7.135 km grid spacing (i.e. based on the 0.1o North Atlantic model resolution at that latitude). Thirty-one vertical levels extend to 3250 m, with 10 to 250 m thicknesses. The domain is a flat-bottomed channel (cyclic in x-direction, walls on y boundaries). The Coriolis parameter f varies with latitude.
A linear equation of state is used, and salinity is constant (35 psu), such that density is proportional to temperature (T).
Temperature profiles for a typical cold eddy core Tin(z) and the mean background Tout(z) were based on density features near 50.87oN 27.75oW from a 10-day average of the 0.1o North Atlantic simulation ending 2 Mar 1994, tranformed to T using the linear equation of state. These T profiles are shown in Fig. 1a.
The initial 3-D temperature and (cyclonic) azimuthal velocity distributions were then defined as:
That is, density is distributed horizontally as a Gaussian with e-folding scale l. Velocity is then determined based on geostrophic and cyclostrophic balance (the first and second terms of the u equation, respectively; see Appendix below for derivation), with the deepest level (3125 m) taken as the level of no motion. The values of alpha, fo and l are based on the 0.1o North Atlantic simulation at this latitude. A transect of velocity at the sea surface is shown in Fig. 1b. The maximum azimuthal velocity occurs at a radius of l/sqrt(2) = 71 km.
Notes: Several feature models were considered. Spall and Robinson (1990, JPO, p 985) was not used because their du/dr is discontinuous, making geostrophic and cyclostrophic balance difficult to compute. Instead, I follow Legg et al. (1998, JPO, p 944) with the modification of, rather than defining u(r,z) and computing T(r,z), the opposite is done, given that we have more robust "observations" of T(z). This makes the derivation somewhat similar to Molemaker and Dijkstra (2000, JPO, p 475), though with the cyclostrophic term included, as in Legg et al. (1998). For this density initialization the cyclostrophic term turns out to be 0-8% of the geostrophic term.
A passive tracer ("SF6") is initialized in the top 104 m "inside" the eddy, as defined by the radius of maximum azimuthal velocity. Biharmonic horizontal diffusivity and viscosity coefficients of -2.262e+9 m4/s and -6.785e+9 m4/s were used, similar to that in the 0.1o North Atlantic simulation at this latitude.
Run 1: 40-day Control run. No surface forcing.
Run 2: 40-day run with surface cooling on days 11-15. Surface cooling is conducted as restoring level 1 T to 4.44 oC (as in the 0.1o North Atlantic simulation at this latitude in March).
Even though W in the initial conditions is near zero (as it should be analytically), by day 1 it has formed a dipole (see movie). This dipole occurs whether the cyclostrophic term is included or not, and whether a cartesian or latitude-dependent grid is used. In any case, it is not unusually strong viz. 1 m/d at its strongest at 1000 m. Most likely, it is due to the "beta spiral" effect, i.e. dw/dz = ßv/f, such that w(z) = -zßv/f, which agrees with its initial pattern and magnitude. So is the W dipole a characteristic of eddies in nature (in the absence of other processes e.g. decay)?
It was first thought the W dipole was related to westard propogation i.e. upwelling on the west side and downwelling in the east in the Eulerian frame. A W of 1 m/d would mean the 100-km radius, 200 m anomaly (Fig. 1a) is moving at 0.5 km/d, which roughly agrees with its mean observed movement of 1 km/d. However in the first 10 days the eddy actually moves N, not W (Fig. 2). Also, as the run progresses, the W dipole shifts orientation to SW-NE, possibly due to the impact of the cyclonic horizontal advection; the eddy then moves towards the NW. Thus the eddy always moves orthogonal to the W pattern, suggesting no real relationship between the main W pattern and Eulerian translation (i.e. one does not appear to be directly influencing the other). As the main W pattern does not reflect the horizontal motion, the W caused by horizontal motion must be small relative to the W pattern caused by the beta-spiral effect. Close inspection of the movie after day 15 does show some W on the outer edges related eddy propogation.
Why does the eddy move N and then NW instead of W (Fig. 2)? Why does the eddy elongate? Movie #2 suggests the W dipole generates density anomalies, seen most clearly in T at 3125 m (which is initially homogeneous). These in turn give rise to a secondary circulation of cyclonic rotation in the western half of the gyre and anticyclonic in the eastern half; in fact below 1000 m by day 40 this "secondary circulation" has grown (fed by W) to dominate the flow field there. This secondary flow field explains why the eddy moves first N and the NW, as the center of mass follows the cyclonic lobe.The eddy moves about 1 km/d, which agrees with the secondary flow speed of 1-2 cm/s. Presumably elongation of the eddy occurs because of the differences between the shallow flow field and the deep flow field.
Max(W), min(W) and mean(abs(W)) increase with time (Fig. 3, and Movie #2). Why? Spindown should cause W to decrease with time, rather than increase; such an initial adjustment occurs on days 1-2. Probably the increase after day 30 is related to the eddy elongation (the elongation actually begins around day 20, but becomes more severe with time). That is, W -> elongation -> higher W -> more elongation-> etc.
Even though mean(W) is zero (not shown), Mean(W*SF6) in Run 1 is negative (Fig. 3d), indicating W is net downward inside the eddy (r < l/sqrt(2)) where SF6 is high. Presumably this small net downwelling inside the eddy of about -0.02 m/day at 104 m is related to eddy decay.
The W dipole allows trace amounts of SF6 to be advected to 1000 in the downwelling lobe. But as W near 500 m is only about 0.1 m/d = 4 m in 40 days, not much SF6 is subducted.
Although the elongation of the eddy was first thought to possibly be a mistake, it appears to be due to W resulting from the beta spiral effect, and thus perhaps it is not undesirable. That is, we possibly could keep the eddy circular by running on an f plane, or by using a barotropic eddy, but these would be less realistic cases. Eddies have a certain life cycle.
With 5 days of surface cooling, which significantly deepens the MLD (as exhibited by SF6 in the movie), the eddy does not go baroclinically unstable. Maybe that is ok; in the 0.1o North Atlantic simulation, the eddies don't really break apart. Run the cooling longer? e.g. 30 or 40 days, as in the 0.1o North Atlantic simulation? In the 0.1o North Atlantic simulation, negative vertical advection does occur in March, when there is cooling (though not necessarily convection) over all of the domain for the entire month. Thus the subduction may not be as influenced by the temporal variability of the buoyancy forcing (which is monthly) as by the spatial variability of the hydrography.
Note mixed-layer depth inside and outside the eddy are similar. Does it make a difference which one is deeper than the other, with regards to the subduction across 104 m? Since the MLDs are > 300 m, what causes W at 104 m at all? There are still horizontal density gradients in the mixed layer, which suggests barotropic (shallow-water) processes. However the deeper eddy hydrography appears to have the strongest influence on W at 104 m. W is maximal at 1000 m, and the W pattern at 104 m closely mimics it (though weaker, as W must go to zero at the sea surface). Convection/MLD deepening does affect W at 104 m (compare W in Fig 4 vs 5), however the primary pattern still follows that of W at 1000 m. The deep eddy hydrographic structure also presumably influences the eddy stability and evolution. Thus, even though we are interested in W at 104 m, and the MLDs here are > 300 m, we apparently cannot simplify the simulation to e.g. a well-mixed cylinder of 6.45 oC within a well-mixed background of 7.75 oC. The depth of the mixed-layer inside versus outside the eddy probably does not affect W at 104 m much in this particular case.
Do winds (which are not included in this run) in nature primarily influence subduction at the base of the ML (and hence 104 m) through horizontal surface convergence pumping water out through the base of the ML? Note however the net subduction occurs primarily during winter not summer, so convection (or deep MLDs) must play a more crucial role than winds. [But perhaps winds are necessary in addition to convection. Run a 0.1o North Atlantic segment without winds? What about surface convergence caused by the large scale circulation caused by the climatological winds, versus the effect of synoptic winds?]
Mean(W*SF6) is significantly more negative during convection (Fig. 3d). This is confirmed in the comparison on day 13 (Figs. 4, 5, 6): during convection W*SF6 is more negative over the entire eddy center (viz. the upwelling lobe is weaker and the downwelling lobe is stronger, and they do not counterbalance each other). Why does convection cause more downwelling in the eddy center? Perhaps the convection causes instability (deviation from geostrophy) which enhances the decay rate. [Though shouldn't the eddy decay rate be governed by the deeper hydrography?] Thus Run 2 suggests more subduction occurs during the convective period rather than post-convective.
This subduction in the eddy center however may not be like what occurs in the 0.1o North Atlantic simulation, where the negative vertical advection appears more frontal in nature and involves higher W of up to O(10 m/d) at 104 m (though these high W dipoles, which also occur in summer, are not necessarily related to the subduction process e.g. AVERT-W10N10). Is subduction in the 0.1 North Atlantic simulation enhanced by eddy-eddy interactions? Initialize the model with more eddies e.g. 3-4 cold and 3-4 warm eddies? Yet the subduction cannot be solely due to eddy-eddy interactions, as it does not happen in summer; convection/MLD deepening or gradients must play the key role. Also in the 0.1o North Atlantic simulation there is a large-scale density gradient in this region viz. warmer waters to the SW and colder waters towards the NE. Does the interaction of eddies with this large-scale, sloped density structure have an impact on subduction processes? I.e. subduction equals the rate at which water is drawn along these surfaces. Initialize a test run with the 0.1o North Atlantic 2 Mar 1994 snapshot exactly? Because if the W structure in the test run is not similar to that in the 0.1o North Atlantic simulation, the subduction process may not be the same. Perhaps Runs 1 and 2 indicate that eddy-eddy interactions (or winds) greatly enhance and restructure W and thus the subduction process. This would need to be demonstrated by runs with eddy-eddy interactions (or winds).
SF6 on level 1 in the movies and in Fig. 4 is low in the center of the eddy because MLD is one level deeper there.
SF6 on level 15 (229 m) in the Run 2 movie increases due to MLD increasing, but shows no obvious signs of subduction. The SF6 on level 17 is elevated over a smaller area, as MLD reaches level 17 only in the very center of the eddy. No significant SF6 reaches level 18 by day 20; if it reaches it later, it is probably due to diffusion rather than subduction. Thus the SF6 distribution shows no obvious signs of subduction.
Possible Future Directions:
1) The generation of the deep velocity structure may not be desirable; eliminate it? by eliminating the W dipole? The elongation (i.e. horizontal density gradients) and W dipole may or may not affect the subduction processes.
2) Does the fact that W increases with time and the eddy elongates indicate that the IC are not stable? If so, what is the instability caused by? How can the IC be improved? Do other terms need to be taken into account in deriving u(r,z), or is a Gaussian T(r,z) just not stable? or is Tin(z) or Tout(z) the problem?
3) Does the eddy's vorticity influence its northward movement? Estimate if PV is conserved. But Rel Vort=dv/dx-du/dy integrated over the domain is zero. (It is positive inside the Umax, but negative in the decellerating zone outside it. [does this agree with the analytical solution?]) Estimate angular momentum = moment around the center of rotation: (x-xo)v-(y-yo)u. But at depth, 2 centers of rotation form...use which one? Compute both; conservation should indicate which one is appropriate. It turns out the center reference point does not matter; angular momentum is not conserved; it decreases then increases again. Probably vorticity is not conserved either. KE (not shown) decreases slightly and then increases slightly; must be conversion from PE to KE during elongation.
Possible New Runs:
1) Run with 30 or 40 days cooling
2) Run with wind forcing
3) Run a 0.1 North Atlantic segment without winds
4) put the AVERT diagnostic back into the code
5) new IC: center the eddy on a U point instead of a T point, to make sure that doesn't make a difference
5) new IC: exactly like Legg to see if a W dipole forms
6) new IC: add more stagnant layers on the bottom, to see if this inhibits the deep counter-rotating eddy from forming
7) are there other IC forms that don't evolve a W dipole? e.g. one that has W just be negative in the center (i.e. decaying eddy)?
8) new IC: put the eddy on an f-plane, so it wont drift and deform? but maybe the deformation intensifies the subduction, or the beta effect is important for subduction processes. Run on a beta plane?
9) the secondary flow is strongest on the bottom level. Does the bottom BC impact whether this secondary circulation is generated or not? I.e. can more (or less) bottom friciton suppress it? If so, is the the proper way to correct the simulation? the natural way?
9) new IC: barotropic eddy; this way W won't cause density anomalies
10) new IC: multiple eddies
11) new IC: large scale background density gradient
12) new IC: use the 2 Mar 1994 snapshot exactly
[re-initialze SF6 after convection (i.e. on day 16)? to show post-convective advective transport across 104 m? But we can already compute W and W*SF6 at 104 m.]
[initialize SF6 like the NO3 distribution i.e. proportional to density i.e. T? But W*T looks like W.]
[Do a run with Laplacian diffusion of 10 m2/s? Do a 2 km-resolution run? Though the "blue spot" in the North Atlantic simulation is not caused by these.]
[initialize the eddy exactly as Legg et al 1998? But as they define velocity as decreasing exponentially with depth, it is impossible to get a deep mixed layer both inside and outside the eddy.]
Fig. 4. Run 2. Convection occurs days 11-15:
Fig. 5. Run 1. No convection:
Fig. 6. Run 2 W - Run 1 W: