20 Hydrostatic balance

Schematic for hydrostatic balance, i.e. a vertical force balance between the gravitational acceleration, g, and the vertical components of the PGF.

We know from experience that pressure increases with depth in the ocean, but why? Consider a water parcel (an infinitesimal control volume) in the ocean interior.  Because pressure is increasing with depth, there will be a greater total force on the lower surface than on the upper surface, associated with a net upward PGF.

Why don’t water parcels accelerate upward in response to the PGF? In the ocean there is a steady-state balance between the vertical component of the PGF and the downward-directed gravitational acceleration (g=9.81 m s^{-2}, the gravitational force per unit mass) that keeps water parcels in the ocean close to a state of zero vertical acceleration.

This balance can be written, with the acceleration (0) on the left, and forces on the right:

    \[ 0=-\frac{1}{\rho}\frac{\partial P}{\partial z}-g. \]

We can thus calculate the vertical pressure gradient

    \[ \frac{\partial P}{\partial z}=-\rho g. \]

Since the right side is always a negative number, pressure decreases as we move upwards (to greater values of z) within the ocean.

By integrating this expression from some depth below the resting ocean surface, z=-H, to the sea surface, z=0, we can find the hydrostatic pressure at depth H.

    \[ \int_{-H}^0 \frac{\partial P}{\partial z} dz=P(0)-P(-H)=-\rho g \left. z \right]_{-H}^{0} . \]

Oceanographers commonly use a convention that the atmospheric pressure P(0)=0, thereby wrapping the standard atmosphere into the background state. The pressure at depth H becomes

    \[ P(-H)=\rho g H. \]

If you are not yet familiar with integration, you can instead think of a finite difference representation of the gradient over a small distance,

    \[  \frac{\partial P}{\partial z} \approx \frac{\Delta P}{\Delta z}=\frac{P(0)-P(-H)}{0-(-H)}=-\rho g, \]

and you will end up with the same expression when you solve for P(-H).

Key Takeaways

Hydrostatic Balance is the state where the gravitational acceleration balances the vertical component of the pressure gradient force.

The value of the hydrostatic pressure at depth H in a fluid (where z=-H) is

    \[ P(-H)=\rho g H. \]

The hydrostatic pressure gradient has only a vertical component, equal to

    \[ \frac{\partial P}{\partial z}=-\rho g. \]

 

 

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