# Physical Interpretation of the Equation of Motion

Ocean dynamics can often be described by the equation of motion $$\frac{D\vec{u}}{Dt}=-\nabla\Phi-\frac{1}{\rho}\nabla p-\vec{f}\times\vec{u}.$$ 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 $$\Phi=-g\hat{z}$$ (where $$\hat{z}$$ is the vertical unit vector), $$\rho$$ is the density, $$p$$ denotes pressure, $$\vec{f}=f\hat{z}$$ (where $$f$$ is the Coriolis parameter $$f=2\Omega\sin(\phi)$$).

• Can you define all the variables in your equation? Ultimately, it's just an expression of $F = ma$, but we can't really explain what all the $F$ terms are (the RHS) without knowing what the variables stand for. Feb 25, 2020 at 2:59
• The equation is the Navier Stokes equation expressed in the rotating reference frame of the Earth, see whoi.edu/cms/files/12.800_Chapter_4_'06_25333.pdf Feb 25, 2020 at 11:17
• @ArthurMorris Your link does not work. Are you able to try again? I would enjoy reading it. Feb 26, 2020 at 4:09
• @Steven This would be on topic at Earth Science Stack Exchange as well Feb 26, 2020 at 16:23
• @tpg2114 Are there any further terms you wish for me to clarify? I have made an edit :) Feb 28, 2020 at 9:01

$$-\nabla\Phi$$ 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 $$-\nabla\Phi = -g\hat{z}$$ 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 $$-\nabla\Phi$$ 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, $$-\overrightarrow{f}\times\overrightarrow{u}$$, represents the Coriolis effect.