8.2 Finite volume budget for the total tracer

The time tendency for the total amount of tracer in a cell is given by

 

$$\partial_{t} [ [T]_k ] [ [T_{t} ]_k ] + [T(\eta) \,\eta_{t}] \, \delta_{k1},$$ = (8.2)

where the square brackets denote volume integration. For example, within the Boussinesq approximation, the volume integral of $T = \rho_{o} \, s$ represents the total mass of salt within this volume. Likewise, the volume integral of $T = \rho_{o} \, c_{p} \, \theta$ represents the total heat (in units of energy) within this volume. For T representing the mass/volume of a passive tracer, the volume integral of T is the total mass of this tracer in the volume. The first term on the right hand side of equation (8.2) arises from the time tendency of the tracer concentration inside the box. The second term occurs only for a surface cell and comes from the time tendency of the surface height multiplied by the surface tracer concentration. This term alters the tracer budget through changing the volume of the box, and it is absent in the rigid lid approximation for which $\eta =0$. Note that the value of the tracer in this expression is that at the ocean side of the free surface $T(\eta)$. Use of the surface kinematic boundary condition (7.16) brings the tracer budget equation (8.2) to the form

 

$$\partial_{t} [ [T]_k ] \left[ [-\nabla \cdot ({\bf u}T + {\bf F} ) ]_k \right] + \left[ ... ...( w + q_{w} - {\bf u}_{h} \cdot \nabla_{h} \eta \right) \right] \, \delta_{k1},$$ = (8.3)

where all the terms multiplying $\delta_{k1}$ are evaluated at $z=\eta$. Integration by parts leads to

$$[ [ \nabla_{h} \cdot ({\bf u}_{h} \, T + {\bf F}_{h} ) ]_k ] [ \nabla_{h} \cdot [ {\bf u}_{h} \, T + {\bf F}_{h} ]_k ] - [ \nabla_{h} \eta \cdot ({\bf u}_{h} \, T + {\bf F}_{h} ) ] \, \delta_{k1},$$ = (8.4)

where again the terms multiplying $\delta_{k1}$ are evaluated at $z=\eta$. Using this result in equation (8.3) brings about a cancellation of the $\nabla_{h} \eta \cdot {\bf u}_{h} \, T$ term appearing in the $\delta_{k1}$ part of the budget. The vertical divergence integrates to

$$[ [ \partial_{z} \, (w \, T + F^{z}) ]_k ] [ (w \, T + F^{z})_{z=z_{k-1}} - (w \, T + F^{z})_{z=z_{k}} ],$$ = (8.5)

which is the divergence of the area integrated flux across the horizontal cell faces. These results render

 

$$\partial_{t} [ [T]_k ] - [ \nabla_{h} \cdot [ {\bf u}_{h} T + {\bf F}_{h} ]_k ] - [ (w \, T + F^{z})_{z=z_{k-1}} - (w \, T + F^{z})_{z=z_{k}} ]$$ =

$$[ T \, (w + q_{w}) + \nabla_{h} \eta \cdot {\bf F}_{h} ] \, \delta_{k1}$$ + (8.6)

For a surface cell with k=1, there is a cancellation of the $T(\eta) \, w(\eta)$ term to yield the budget

 

$$\partial_{t} [ [T]_{k=1} ] - [ \nabla_{h} \cdot [ {\bf u}_{h} \, T + {\bf F}_{h} ]_k ] + [(w... ...{z=z_{1}} + (q_{w} \, T - F^{z} + \nabla_{h} \eta \cdot {\bf F}_{h})_{z=\eta} ]$$ = (8.7)