Physical Interpretation of the Equation of Motion

Physics Asked on May 8, 2021

Ocean dynamics can often be described by the equation of motion
$$frac{Dvec{u}}{Dt}=-nablaPhi-frac{1}{rho}nabla p-vec{f}timesvec{u}.$$
I am searching for a physical meaning of this equation, to help my understanding. I particularly would like to understanding the physical meaning of the terms on the RHS of the above equation. Here, $D/Dt$ denotes the material derivative.

Here, we denote $Phi=-ghat{z}$ (where $hat{z}$ is the vertical unit vector), $rho$ is the density, $p$ denotes pressure, $vec{f}=fhat{z}$ (where $f$ is the Coriolis parameter $f=2Omegasin(phi)$).

One Answer

I stumbled upon this question while searching for something else. You have probably found and answer yourself by now, but here's a quick attempt to answer your question anyway.

$-nablaPhi$ represents gravity. $Phi$ is the so called geopotential, which is a nice way to handle gravity. If we want the simple, high-school physics version of gravity where we have a constant acceleration $g$ pointing downwards, we let $$Phi = gz$$ (note, that's an ordinary $z$, not the unit vector!) and thus get $$-nablaPhi = -ghat{z}$$ But we can also use more complicated geopotentials, taking irregularities in the earth's mass distribution into account. We can also include the effect of the centrifugal force, due to the earth rotating, in the geopotential, which would make $-nablaPhi$ represent the apparent gravity rather than the actual gravity.

$-frac{1}{rho}nabla p$ is the so called pressure gradient force (per unit mass). It is always directed towards lower pressure and represent the fact that high pressure regions will tend to "push" water towards lower pressure. This is very intuitive, and how you would expect fluids to behave.

The last term, $-overrightarrow{f}timesoverrightarrow{u}$, represents the Coriolis effect.

Answered by blupp on May 8, 2021

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