# 10.19.10: Relationship between Lagrangian mean, Eulerian mean and Stokes drift with dissipation¶

I show here the relationship between the Lagrangian mean and Eulerian mean velocity, Stokes drift and the coefficient of Laplacian dissipation. Figs. 1 and 2 show the typical pattern of the Lagrangian-mean, Eulerian-mean velocities and Stokes drift in one experiment. The Stokes drift always cancel the Eulerian-mean flow so that the Lagrangian-mean flow, that has the same pattern as the Eulerian-mean flow, is nonetheless weaker than the latter.

Figure 1: Zonal (a) Eulerian-mean velocity, (b) Stokes drift (calculated at the third order from the wave field), (c) sum of (a) and (b) and (d) Lagrangian-mean velocity. All quantities have been computed for the days 2125-2225. The Laplacian dissipation coefficient is KH=1000 m2/s and the unitless amplitude of the forcing is 500 (exp16).

Figure 2: Same as Fig. 1 but for the meridional velocities.

Figs. 3 and 4 show the evolution of some measure of each of these mean velocity components to the coefficient of Laplacian dissipation. Although all components vary, the evolution of the Eulerian-mean flow is qualitatively the same as that of the Lagrangian-mean flow, as long as the coefficient of dissipation (and dissipation) is not too small. Thus, although for weaker coefficient, both the Stokes drift and the Eulerian-mean flow increases, the increase of the latter is still larger than the increase in the former so that the difference, the Lagrangian-mean flow, evolves in the same direction as the Eulerian-mean flow.

Finally, notice that the theoretical prediction of the Stokes drift –valid at the third order in wave amplitude (the dashed red lines)– is not good for the meridional component but is relatively good for the zonal component. This is important as it means we could a priori estimate the zonal Lagrangian flow in observations from Eulerian time series alone.

Figure 3: A measure of the Eulerian-mean and Lagrangian-mean velocities and Stokes drift versus the coefficient of Laplacian dissipation in the suite of experiments for which the forcing amplitude is 1000 (unitless): (a) zonal components and (b) meridional components. The estimate of the Stokes drift in plain red line is from the difference between the Lagrangian-mean and the Eulerian-mean velocities and is thus the “true” estimate. The estimate in red dashed line was computed using the theoretical Stokes formula and the wave field, valid at the third order in wave amplitude. The estimate of the Lagrangian-mean velocity in plain black line is from particle trajectories and is thus the “true” estimate. The estimate in black dashed line is from the sum of the Eulerian-mean component and the Stokes drift computed theoretically. The measure for the zonal component are the spatial averaged between 5°E-15°E and 29°N-31°N. The measure for the zonal component are the spatial averaged between 5°E-15°E and 27°N-29°N

Figure 4: As in Fig. 3 but for a suite of experiments with twice the forcing amplitude.

computed with theory_test_script.m in the respective directories of RESEARCH/MODELISATION/HIM/studies/diss_train_of_eddies/exp2/*/analysis_1d/ and RESEARCH/MODELISATION/HIM/studies/PV_and_dissipation/forced_damped_wave/. The matlab routine to produce Figs. 3 and 4 is plot_synthesis_Eulerian_Lagrangian_Stokes_diss.m in RESEARCH/MODELISATION/HIM/studies/PV_and_dissipation/forced_damped_wave/exp15.