January 13, 2023
Journal Article

## Abstract

Tracer variance budgets can be used to estimate bulk mixing in a control volume. For example, simple, analytical, bulk formulations of salt mixing, defined here as the destruction of salinity variance, can be found for estuaries with a riverine source of fresh water and a 2-layer exchange flow at the mouth using salinity as a representative tracer. For a steady case, the bulk salt mixing, $M$, can be calculated as $M=Q_R S_{out}^2 + Q_{in} (S_{out}-S_{in})^2$, where $S_{in}$ and $S_{out}$ are the representative salinities in the estuarine exchange flow, and $Q_R$ and $Q_{in}$ are the river and landward volume fluxes, respectively. $M$ can be considered as the sum of mixing pathways, where each pathway has a mixing of $Q\,\Delta S^2$, where $Q$ is the volume transport and $\Delta S$ is the salinity difference across the pathway. For the estuary case, one mixing path is associated with the river inflow, the other is associated with the inflow of salty, oceanic water. This concept of linking mixing to input-output pathways is extended, in simple box models, from estuaries to scenarios with an arbitrary number of inputs/outputs, as might be found in a complex estuarine network, in a region on a continental shelf, or any other control volume with multiple exchanges. The box model with 3 inputs and 2 outputs is tested in an idealized 3D model river plume. This approach allows for the estimation of mixing in a control volume with each the relative contributions of each input and output to the bulk mixing quantified.

Published: January 13, 2023

## Citation

Qu L., R.D. Hetland, and D. Schlichting. 2022. Mixing pathways in simple box models. Journal of Physical Oceanography 52, no. 11:2761–2772. PNNL-SA-171678. doi:10.1175/JPO-D-22-0074.1