Some thoughts about tracer evolution¶

Case when the tracer concentration N is conserved following parcels¶

Water parcels have to follow contours of N, even if those move in time.
At a fixed point, the only change in N is due to the flow that is perpendicular to the N-contours.
If the N-contours are stationary, then the flow is stationary and along N-contours.
The inverse is also true: If the flow is everywhere and all the time parallel to the N-contours, then both the flow and the N-contours are stationary. Demonstration #1: If at time t, the flow is parallel to N-contours everywhere, then at time t+dt (where dt<<1), the map of N is the same as at time t. Because the flow at t+dt is again parallel to the N-contours, the map at t+2dt is still going to be same so that the map and the contours of N are stationary in time. According to the previous statement, the flow is then also stationary. Demonstration #2: If the N-contours are not stationary, that means they have been advected by a velocity perpendicular to the N-contours which is contradictory to the initial hypothesis. The N-contours ae thus stationary.
Thus, N-contours are stationary if and only if the flow is parallel to the N-contours all the time.

If the N-contours and M-contours are at some point in time parallel to each other, then they are parallel all the time.

The wave has a period Tw and we define the time-mean N-contours as the N-contours averaged over a wave cycle.

First, the case of the wave propagating zonally along a zonal channel: the wave may have a Stokes drift that should be zonal given the symmetry of the problem. Parcels are thus on averaged advected zonally but not meridionally, so that the time-mean N-contours should be stationary.
Inversely, can we say that if the time-mean N-contours are stationary, then the Lagrangian time-mean flow has to be along the time-mean N-contours?
Can we then extend all the results above to the case of the time-mean N-contours and Lagrangian time-mean flow? That are: - The time-mean position of water parcels follow the time-mean N-contours. - At a fixed point, the only change in the time-mean N is due to the Lagrangian time-mean flow perpendicular to the time-mean N-contours. - The time-mean N-contours are stationary if and only if the Lagrangian time-mean flow is parallel to the N-contours all the time. - If the time-mean contours of N and M are on some period of averaging parallel to each other, then they are parallel all the time.

If the eddies are advecting N across the time-mean N-contours and these N anomaly are mixed with their surrounding, then the gradient of N weakens with time and the time-mean N-contours are not stationary over a long time scale.
To keep the time-mean N-contours stationary, you need a source and sink of N at the right places to counteract the mixing effect of the eddies.