# 10.15.10: Relationship between Lagrangian mean velocity across mean PV contours and dissipation - Part 2¶

Two suites of numerical experiments are used to study the relationship between the mass transport across Lagrangian-mean PV contours and the explicit dissipation acting in the model. The second suite of experiments have the same characteristics as the first one, except the forcing amplitude is doubled. Both have Laplacian dissipation (of coefficient KH).

The various Lagrangian-mean quantities are qualitatively the same in all experiments. An example is shown in Fig. 1. The Lagrangian-mean flow is composed of three zonal jets, a westward one centered in the middle of the domain and an eastward jet on both sides (not shown). The two circulations are closed within the western boundary layer and in the middle of the basin with a convergent quasi-meridional Lagrangian-mean flow (Fig. 1b). Besides the overall amplitude of the circulation, the main difference between all experiments is the relative amplitude of the time rate of change of Lagrangian-mean PV (panel a) compared to the Lagrangian-mean velocity across Lagrangian-mean PV contours (panel b). This is discussed further below.

Figure 1: (a): Time rate of change the Lagrangian-mean PV.(b): Lagrangian mean velocity across the Lagrangian-mean PV contours. (c); Sum of (a) and (b). (d): Lagrangian-mean velocity across mean PV contours deduced from the change of PV following water parcels. (e) Lagrangian-mean velocity across mean PV contours deduced from dissipation along parcel trajectories. 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).

The strength of the Lagrangian-mean circulation is measured by the maximum amplitude of the zonally-averaged (between 5°E and 15°E) Lagrangian-mean velocity across Lagrangian-mean PV contours between 27°N and 29°N (panel b in Fig. 1). The strength of the dissipation is measured similarly using the Lagrangian-mean dissipation of Fig. 1e. The relationship between the two quantities and between the dissipation and its Laplacian coefficient are shown in Fig. 2.

Figure 2: (a): Lagrangian-mean dissipation (panel e in Fig. 1) versus Lagrangian-mean velocity across mean PV contours (panel b in Fig. 1). (b): Lagrangian-mean dissipation versus its Laplacian coefficient. The suite of experiments in red has twice the amplitude of the suite of experiments in blue. For the “blue” suite, the open symbol is used to relate the two blue curves. All experiments in this suite are run with a baroclinic time step of 1200 s. For the “red” suite, squares are used for experiments run with a baroclinic time step of 1200 s, circles for a baroclinic time step of 800 s; this difference enables to relate the two red curves. The dashed line in a shows the one-to-one relationship between the transport and dissipation in the case of perfect stationarity.

The first overall relationship: the Lagrangian-mean velocity increases with dissipation (Fig. 2a). This is expected from theory but it is the first time, to my knowledge, that this relationship is explicitly shown. The second relationship: dissipation first increases with the Laplacian coefficient and then decreases as velocity gradients weaken faster than the coefficient increases. Thus, when the coefficient is increased, the transport first increases then decreases (Fig. 2a); say differently, for a given forcing amplitude, there is an upper bound for the transport across mean PV contours, whatever the value chosen for the dissipation coefficient.

With weaker and weaker coefficient, the model is less and less in a stationary state and the relative amplitude of the time rate of change of the Lagrangian-mean PV contours (Fig. 1a) increases; this results in the marks in Fig. 2a to move away from the one-to-one relationship (dashed line) between the transport and dissipation valid, in theory, for the perfectly stationary state.

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 Fig. 2 is plot_synthesis_transport_vs_diss in RESEARCH/MODELISATION/HIM/studies/PV_and_dissipation/forced_damped_wave/exp15.