7.3.3 The linearized free surface height equation

Recall the two equivalent equations for the free surface height, repeated here for convenience

$$\eta_t - \nabla_{h} \cdot {\bf U} + q_{w}$$ =

$$w(\eta) - {\bf u}_{h}(\eta) \cdot \nabla_{h} \eta + q_{w}.$$ = (7.37)

Given knowledge of ${\bf U}_0$ rather than ${\bf U}$, it is possible to determine a linearized free surface height

 

$$\eta^{0}_{t} = - \nabla_{h} \cdot {\bf U}_{0} + q_{w}.$$ (7.38)

As a brief aside, it is useful to note that if one vertically integrates the continuity equation between -H and 0, then uses the bottom kinematic boundary condition (7.15), one finds that

 

$$w(z=0) = -\nabla_{h} \cdot {\bf U}_{0}.$$ (7.39)

Hence, equation (7.38) can be equivalently written

 

$$\eta^{0}_{t} = w(0) + q_{w}.$$ (7.40)

Considering this result as a diagnostic for w(0), one sees that nonzero w(0) arises from the difference between the free surface height tendency $\eta^{0}_t$ and the fresh water flux q_w. For example, when there is zero fresh water flux, $w(0) \approx \eta^{0}_t$, whereas in a steady state, w(0) arises just from the input of fresh water. Note that a positive input of fresh water into the ocean model, with q_w > 0, drives a negative vertical velocity w(0) < 0. Of note is the linear nature of equation (7.40) for $\eta^{0}$. This property should be contrasted with the nonlinear kinematic boundary condition satisfied by $\eta$. To pursue this point a bit further, start from the surface kinematic boundary condition and write

$$(\partial_{t} + {\bf u}_{h}(\eta) \cdot \nabla_{h} )\eta w(\eta) + q_{w}$$ =

$$\left( w(0) + \int^{\eta}_{0} dz \, w_{z} \right) + q_{w}$$ =

$$\eta^{0}_{t} + \int^{\eta}_{0} dz \, w_{z}.$$ = (7.41)

Hence, neglect of the advection of free surface height, as well as the vertical shear of the vertical velocity within the layer between $\eta$ and 0, brings $\eta \rightarrow \eta^{0}$. Otherwise, the two heights differ. That is, a linearized version of the equation satisfied by $\eta$ yields the equation satisfied by $\eta^{0}$.