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The Boussinesq approximation
The x3 momentum equation reads (after neglecting rotation):
i.e., part of the pressure is associated with offsetting the weight of the fluid above.
We can subtract out a significant part of this as follows:
0 3Let ' ,x x t
Reference density
Background density variation – exists in the absence of motion
Perturbation density – association with motion
The preferred ordering (which is often valid in oceans, estuaries and lakes) is 0 3 ' ,x x t
Likewise we write the pressure as 0 3 ' ,p p x p x t
such that in the absence of motion
00
3
0p
gx
If p0 is defined by this eqn., then we can subtract out the background hydrostatic pressure gradient and the weight force associated with the density field that exists in the absence of motion.
We can use the ordering of the density field to make an important simplification/approximation:
0 0'Du Du
Dt Dt
i.e., the mass of each fluid particle that determines what acceleration results from a given force is approximately constant. On the other hand we retain the effect of density variations in the buoyancy (gravity) term (’ g). This requires that
' 'Du
gDt
Dug
Dt
Particle accelerations << g
This approximation is known as the Boussinesq approximation
If '<<, we require that
Navier-Stokes equation with the Boussinesq approximation
We also need to make the same approximation in the mass-conservation equation, i.e.
which implies that, as a consequence of the Boussinesq approximation,
Note that we assumed this a priori when writing the viscous term as given above...
01
Dt
D
The difference between incompressibility and the Boussinesq approximation
If a flow is incompressible, this implies that the density following a fluid particle is identically zero, which gives the equations for conservation of momentum and mass as
Under the Boussinesq approximation, the density following a fluid particle is not constant, but its time rate of change is much smaller than that due to changes resulting from velocity gradients. This enables one to write the momentum and mass conservation equations as
The Boussinesq approximation does two things: it linearizes the acceleration term in the Navier-Stokes equations and enables use of the continuity equation while retaining the effects of density in the momentum equation.
Incompressible
Boussinesq
How do we cope with free surfaces?
1 2, ,x x tx3=0
x3
-x3
From before, we had p=p0+p', where p0 was the pressure fieldin the absence of motion, while p' was that associated with motion.We can define an alternate splitting of the pressure as p=ph+0, where:
ph = Hydrostatic pressure arising from weight of fluid(can include motion this time)
0= Dynamic, or nonhydrostatic, pressure arising from fluid motion
Defining the hydrostatic pressure as satisfying the balance
we can integrate both sides to obtain ph:
Surface pressurePressure due to depth and free surface: BAROTROPICPRESSURE
Pressure due to density variations:BAROCLINICPRESSURE
Adopting very commonly-used shorthand notation for the horizontalgradient, such that
we have
Surface pressure gradienti.e. Atmospheric pressure.
Barotropic pressure gradient due to free-surface gradient.
Baroclinic pressure gradient due to density gradient.
Substitution into the Navier-Stokes equation with the Boussinesqapproximation gives
Or, component-wise:
Why does water level go down when atmospheric pressure goes up?
An example from SF Bay/Delta (observations):
10 cm (water)
15 cm