# What are baroclinic and barotropic waves?

I have a basic understanding of what baroclinic and barotropic mean but cannot seem to get my head around baroclinic waves and barotropic waves.

• There is not really a difference, they are both waves in the end. Commented Jan 4, 2016 at 10:05
• You need to have a good understanding of stratification for it to make sense. Imagine the surface of the ocean being an extreme case of density difference (air vs water) compared to the interior of the ocean (warmer water vs cooler water). Commented Jan 4, 2016 at 10:18

Barotropic flows are those for which the fluid density $\rho$ depends on pressure $p$ only, i.e. $\rho = \rho(p)$. For example, constant density, isothermal, and isentropic flows are barotropic. Baroclinic flows are flows that are not barotropic. In this case the fluid density depends on pressure AND temperature, salinity, constituent concentration. Heavy fluid sitting adjacent to lighter fluid will sink and rise, respectively.

These flows are said to be baroclinic, meaning pressure breaking.

Consider the two fluid elements below. In the barotropic case, lines of constant $p$ are parallel to lines of constant $\rho$ so there is no mechanism to generate vorticity. On the other hand, in the baroclinic case, the lines of constant $p$ and $\rho$ are not parallel. The latter arrangement give rise to changes in vorticity $\vec{\omega}$ and circulation because the net pressure force no longer passes through the center of mass of the fluid element.

Seen another way, take a look at the vorticity equation with density variation:

$$\frac{D\vec{\omega}}{Dt} = (\vec{\omega}\cdot \nabla)\vec{u} + \frac{1}{\rho^2}(\nabla \rho \times \nabla p)$$

The cross term on the right hand side is the baroclinic torque. When it is zero the flow is barotropic whereas if it is nonzero the flow is baroclinic.

When applied to waves in the Ocean the result is that barotropic relates to waves propagating at the ocean surface. Baroclinic relates to waves propagating along density surfaces in the interior of the ocean (or internal waves).

One good example is the surface tide and the internal tide. As the surface tide (regular tide) sweeps around the main oceanic basins, it rubs against the ocean floor giving rise to internal tides. The former are called barotropic tides, whereas the latter are called baroclinic tides.

References:

P. Kundu, Fluid Mechanics

B. Sutherland, Internal Gravity waves

• -1 Can you go into more detail about what barotropic and baroclinic actually mean? Your answer is not strictly correct - there exist internal waves that are barotropic. Commented Jan 4, 2016 at 15:34
• I was trying to keep it simple but I can put more detail in. Can you give me an example of a barotropic internal wave in the ocean? A quick google search gives zero results for "barotropic internal wave". Commented Jan 4, 2016 at 20:53
• Yes, an internal shallow-water gravity wave is a barotropic wave with essentially reduced gravity. That's not the real ocean - I am playing devil's advocate here. Baroclinicity requires a horizontal gradient in either temperature or salinity. I hope your answer can expand in that direction and will happily +1 then. :) Commented Jan 4, 2016 at 21:19
• I thought that is what you meant. I will put more detail soon. Thanks for setting me straight. Commented Jan 4, 2016 at 21:23
• since the atmosphere tag has also been added will the answer cover atmospheric barotropic and baroclinic waves as well?
– user1066
Commented Jan 5, 2016 at 11:08

The definition of barotropic is that density depends only on pressure. The definition of baroclinic is that density can depend on other variables. This in itself is not particularly useful for understanding what baroclinic and barotropic flow means.

In the context of the ocean, water is pretty much incompressible, so density will be virtually constant for a barotropic ocean. Hence there is no stratification for a barotropic ocean, and the only waves the fluid can support are at the interface with the air. On the other hand, if the fluid is baroclinic (density depending on non-constant temperature, and also salinity), then there can be internal gravity waves caused by a change in density in the z-direction.

For the atmosphere, air is relatively compressible. Hence if the atmosphere is barotropic, it means that the only density changes are due to pressure. Hence temperature is horizontally constant and there is no thermal wind shear in the vertical. This is a more reasonable assumption in the tropics than in the midlatitudes.

• welcome to ESSE. The text looks very similar to Wiki's definition of barotropic fluid. So some references would help
– user1066
Commented Oct 25, 2017 at 15:55