7.1 Hydrostatic pressure with the free surface

To separate the fast barotropic gravity waves from the much slower baroclinic fluctuations, it is necessary to split the hydrostatic pressure into three terms. To start, integrate the hydrostatic equation (4.4) from the atmosphere-ocean interface at $z=\eta(\lambda,\phi,t)$ to some ocean depth z

$$p(\lambda,\phi,z,t) = p_{a}(\lambda,\phi,t) + g \, \int_{z}^{\eta(\lambda,\phi,t)} \rho(\lambda,\phi,z',t) \; dz'$$ (7.1)

In this expression, $\eta$ is the free surface displacement with respect to a resting ocean ($\eta$ can be positive or negative), p_a is the atmospheric pressure at the surface of the ocean, and $\rho$ is the in situ ocean density. Note that p_a and $\eta$ are functions only of the horizontal position $(\lambda,\phi)$ and time. The use of Leibnitz's Rule (equation (4.65)) leads to the expression for the horizontal pressure gradient

 

$$\nabla_{h} p \nabla_{h} p_{a} + g \, \rho(z=\eta) \, \nabla_{h} \eta + g \, \int_{z}^{\eta} \nabla_{h} \rho \, dz'.$$ = (7.2)

This gradient is needed in the horizontal momentum equations. In this expression, the first term arises from horizontal gradients of the atmospheric pressure, the second term from horizontal gradients of the free surface height, and the third term from the vertically integrated baroclinicity in the ocean column above the depth z. It is the gradient of the free surface height which is responsible for fast gravity waves, and so it must be separated from the baroclinic model part. Now consider the integral

$$\int_{z}^{\eta} \nabla_{h} \rho \, dz' \int_{0}^{\eta} \nabla_{h} \rho \, dz' + \int_{z}^{0} \nabla_{h} \rho \, dz'.$$ = (7.3)

The first integral is over the very small vertical distance from the resting ocean height z=0 to the free surface height $z=\eta$, with the free surface height $\vert\eta\vert \approx 100-200\,\mbox{cm}$ at the most. In this region, the ocean has very small vertical density gradients due to the strong mixing effects from interactions with the atmosphere. Therefore, it is quite accurate to assume that

$$\rho(z=\eta) \approx \rho(z=0).$$ (7.4)

This approximation leads to the expression

$$g \, \int_{z}^{\eta} \nabla_{h} \rho \, dz' \approx g \, \eta \, \nabla_{h} \rho(z=0) + g \, \int_{z}^{0} \nabla_{h} \rho \, dz'$$

$$g \, \eta \, \nabla_{h} \rho(0) + \nabla_{h} p_{b},$$ = (7.5)

where

 

$$p_{b}(\lambda,\phi,z,t) = g \, \int_{z}^{0} \rho(\lambda,\phi,z',t) \, dz'$$ (7.6)

defines the hydrostatic pressure field associated with density in the vertical column between z and a resting ocean surface z=0. In turn, the horizontal gradient of this field, $\nabla_{h} p_{b} = g \, \int_{z}^{0} \nabla_{h} \rho \, dz'$, arises from baroclinic effects in that part of the ocean between the resting ocean surface and the depth z. It is for this reason that p_b is often termed the baroclinic pressure field. Note that the full depth integral of p_b does not vanish, as may mistakenly be construed by the adjective ``baroclinic.''

As a result of the well mixed assumption, the horizontal pressure gradient can be written

$$\nabla_{h} p \nabla_{h} \, (p_{a} + p_{b}) + g \, \nabla_{h} \, [ \eta \, \rho(z=0)]$$ =

$$\nabla_{h} \, ( p_{a} + p_{b} + p_{s} ).$$ = (7.7)

In this expression, the surface pressure

$$p_{s}(\lambda,\phi,t) = g \, \rho(z=0) \, \eta,$$ (7.8)

was introduced. This is the hydrostatic pressure head associated with the surface height, where again is it assumed that the density field is well mixed between z=0 and $z=\eta$. The total hydrostatic pressure field has been written

 

p = p_a + p_b + p_s. (7.9)

For z<0 the depth dependent baroclinic pressure is separated from the depth independent atmospheric and surface pressures. Such a separation is important for the methods described below for integrating the momentum equations. Note that for z>0 the baroclinic pressure p_b cancels partly the surface pressure p_s. Thus, for $z=\eta$ the boundary condition $p(z=\eta)= p_a$ is correctly recovered.

It is interesting to make a connection between the prognostic surface pressure used in the free surface method to the diagnostic lid pressure associated with the rigid lid method. To do so, recall that the pressure at an arbitrary depth in the rigid lid method (Section 6.3) can be written

p_rl = p_a + p_l + p_b, (7.10)

where the atmospheric and baroclinic pressures are identical to those used in the free surface. Hence, the lid pressure p_l can be identified with the surface pressure. Indeed, if the rigid lid is allowed to freely deform, the ocean surface would then take the shape given by $\eta$. In practice, the gradient of the surface pressure $\nabla_{h} \, [ \eta \, \rho(z=0)]$ is often approximated (e.g., Killworth et al., Dukowicz and Smith ) by

$$\nabla_{h} \, [ \eta \, \rho(z=0)] \approx \rho_{o} \, \nabla_{h} \, \eta,$$ (7.11)

where the space-time dependent surface density is approximated as a constant

 

$$\rho(\lambda,\phi,z=0,t) \approx \rho_{o} = 1.035\,\mbox{g}\,\mbox{cm}^{-3}.$$ (7.12)

For convenience, this constant is also the constant value for the Boussinesq density used in MOM. The approximation (7.12) is valid so long as $\eta \, \Delta \rho / (\rho \, \Delta \eta) << 1$, where $\Delta \rho$ and $\Delta \eta$ represent typical horizontal variations. This assumption is true in the ocean over the horizontal scales of a model grid box (order 100km). Note that the global averaged annual mean surface density from Levitus (1982) is $1.024\,\mbox{g}\,\mbox{cm}^{-3}$, which is close to the chosen $\rho_{o}$. However, setting $\rho(z=0) = 1.035\,\mbox{g}\,\mbox{cm}^{-3}$ may yield an unacceptably large systematic error for regions where the average density differs drastically from $1.035\,\mbox{g}\,\mbox{cm}^{-3}$. In such cases, it may be more appropriate to use a different value than $\rho_{o}$, or it might be best to avoid the approximation (7.12). Since Summer 1999, the standard explicit free surface in MOM does not use this approximation. Now that the pressure field has been suitably split into the surface and atmospheric pressure p_s+p_a, whose fluctuations are associated with fast barotropic processes, and the baroclinic pressure p_b, whose fluctuations are associated with the slower baroclinic processes, it is appropriate to consider the formulation of the barotropic and baroclinic systems. This formulation provides the focus for the next few sections.