7.4.3 Revisiting the surface stress

The surface stress applied at z=0 for the linearized free surface

 

$${\bf\tau}_{surf}/\rho_{o} \approx q_{w}\, {\bf u}(z=0) + \kappa_m {\bf u}_z\vert _{z=0}$$ (7.60)

was found through taking the $\eta \rightarrow 0$ limit of the $\eta \ne 0$ results. The following discussion attempts to further illustrate this limit. For this purpose, subdivide the surface box in the $\eta \ne 0$ case into two boxes. The first box, with z₁<z<0, is explicitly resolved by the ocean model. It represents the surface model grid cell, and it has a fixed volume. The second layer, between z=0 and $z=\eta$, is considered a virtual model cell. It is not explicitly part of the ocean model, and it has a variable volume. When $\eta <0$, the virtual box overlaps with the z₁<z<0 box, whereas for $\eta >0$ it is distinct from the z₁<z<0 box. Without an extra prognostic equations integrated by the ocean model for the virtual box, the velocity and the tracer concentration in both boxes must be taken to be same. This assumption should be valid since the small virtual box is typically well mixed with the model box due to interactions with the atmosphere. The momentum flux entering the model domain from the virtual box through the level z=0 is given by

 

$${\bf F}_{z_o} (\kappa_m{\bf u}_z)_{z=0} - (w(z){\bf u}(z))_{z=0}.$$ = (7.61)

Again, the first term is the turbulent momentum flux, and the second term is an advective flux representing the vertical advection of horizontal momentum. The goal is to incorporate the effects of the virtual box on the resolved box through an appropriate form of the flux $(\kappa_m{\bf u}_z)_{z=0}.$

The momentum balance of the virtual upper box can be found from the volume averaged velocity equations by replacing z₁ by z₀=0. Resolving the momentum equations for the top model box for ${\bf F}_{z_o}$, taking the limit as $\eta \rightarrow 0$, and neglecting gradients of $\eta$, an approximation for the open vertical boundary condition at z=0 is found to be

$${\bf F}_{z_o} \approx {\bf\tau}_{surf}/\rho_{o} -{\bf u}(z=0) \left( w(z=0) + q_w \right).$$ (7.62)

Combined with equation (7.62), this result can be rearranged to the form

 

$$(\kappa_m{\bf u}_z)_{z=0} \approx {\bf\tau}_{surf}/\rho_{o} - {\bf u}(z=0) \, q_w.$$ (7.63)

This result is identical to the expression (7.61) obtained through the formal $\eta \rightarrow 0$ limit.