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10.08.10: Relationship between Lagrangian mean velocity across mean PV contours and dissipation

I show here the various component of the Lagrangian-mean PV equation in terms of velocity for 4 different simulations that vary in dissipation coefficient. The name and characteristics of each experiment are listed in Tab. 1.

  Baroclinic time step Laplacian viscosity (m2/s)
exp2_w 1200 (SPLIT) 500
exp2_w3 1200 (SPLIT) 250
exp2_w4 1200 (SPLIT) 100
exp2_w5 1200 (SPLIT) 50

** Table 1:** Time step and Laplacian viscosity in the 4 experiments considered.

Each figure of Figs. 1 to 4 below is composed of 5 panels. Panel (a) shows the time rate of change the Lagrangian mean PV. Panel (b) shows the Lagrangian mean velocity across the mean PV contours. Panel (c) is the sum of (a) and (b) and should be equal to either panel (d) or (e) if GLM theory is right. Panel (d) is the Lagrangian mean velocity deduced from the change of PV following water parcels and panel (e) is the Lagrangian mean velocity deduced from dissipation. All quantities have been computed for the days 2125-2225.

A synthesis is shown in Figs. 5 and 6.

Results

  • exp2_w5 is not in a stationary state as the time rate of change of Lagrangian mean PV is relatively large (Fig. 4a). This means that mean PV contours are changing with time and this simulation should not be considered –for now. Although this is also true but to a less extent for exp2_w4, we will still consider this simulation.
  • The velocity deduced from the dissipation (or the change of PV following parcels) is always weaker than the actual Lagrangian mean velocity across mean PV contours (Figs. 5 and 6a).
  • For the three simulations that we should consider (exp2_w, exp2_w3 and exp2_w4), the Lagrangian mean velocity across mean PV contours does increase with dissipation (Fig. 6a). A note of caution, however, is that the increase is relatively small, and is as large as the ‘velocity’ associated with the time rate of change of the Lagrangian mean PV contours. It is possible that it is also as large as the difference in quantities we would obtain if a different baroclinic time step may have been used. Indeed, the same quantities have been computed from another simulation, exp_13, similar to exp2_w except that it is run with a 30-s baroclinic time step. Although the days over which they were computed were different –days 530 to 630–, the difference with the quantities of exp2_w are as large as the difference we see between the simulations due to only to the coefficient of dissipation.
  • Another interesting result is that the dissipation is a monotonic function of the coefficient of dissipation (Fig. 6b), although not a linear one.

Is the next step to explore stronger dissipation 1) to be closer to a stationary state and reduce the misfit due to the change in Lagrangian mean PV contours and 2) to see if the relationship seen in Fig. 6a keeps going or not?

exp2_w

../../../../../_images/theory_test_exp2_w_d2125_22251.png

Figure 1

exp2_w3

../../../../../_images/theory_test_exp2_w3_d2125_22251.png

Figure 2

exp2_w4

../../../../../_images/theory_test_exp2_w4_d2125_22251.png

Figure 3

exp2_w5

../../../../../_images/theory_test_exp2_w5_d2125_22251.png

Figure 4

Synthesis

../../../../../_images/synthesis.png

Figure 5: Zonal average between 5°E and 15°E of (upper) Lagrangian mean velocity across the mean PV contours (panels (b) in Figs. 1 to 4), (middle) Lagrangian mean velocity deduced from the change of PV following water parcels (panels (d) in Figs. 1 to 4) and (lower) the Lagrangian mean velocity deduced from dissipationand (panels (e) in Figs. 1 to 4).

../../../../../_images/synthesis_2.png

Figure 6: (a) On the vertical axis, maximum amplitude of the Lagrangian mean velocity of Fig. 5a between 27°N and 29°N versus, on the horizontal axis, that of the Lagrangian mean velocity of Fig. 5b. (b) On the vertical axis, coefficient of Laplacian dissipation versus, on the horizontal axis, the maximum amplitude of the Lagrangian mean velocity of Fig. 5b.


computed with theory_test_script.m in the respective directories of RESEARCH/MODELISATION/HIM/studies/diss_train_of_eddies/exp2/*/analysis_1d/. The matlab files to reproduce Fig. 5 are found in these directories and its name is VC_zav_5E15E_*.mat.