OEAS 604: Introduction to Physical Oceanography

Conservation of MassChapter 4 – Knauss

Chapter 5 – Talley et al.

1

Outline• Cartesian coordinate system• Conservation of mass• Derivation of continuity equation• Eulerian and Lagrangian reference frames• Boussinesq Approximation• Incompressible form of continuity equation• Application of continuity

2

z

x

y

Cartesian Coordinate System

The location of any point in space can be uniquely described by its coordinates (x,y,z)

Similarly the velocity vector of an object can be uniquely described in Cartesian Coordinates

Typically use u to denote a vector in the x-directionv to denote a vector in the y-directionw to denote a vector in the z-direction

3

Conservation of Masschange in mass = flux in – flux out

ΔxΔyΔz

mass = density × volume

change in mass = flux in

flux in = flux out

1. If diameter of pipe is 10 m2 and the velocity of water through the pipe is 1 m/s, what is the flux in?

2. How fast is volume changing?3. How fast is water rising?

4

Conservation of Mass in a Cartesian Coordinate System

ΔxΔy

Δz

If box is filled with water of density ρ, what is the mass of water in the box?

Mass = ρ × Δx × Δy × Δz

ρ

5

Conservation of Masschange in mass = flux in – flux out

ΔxΔy

Δzu1ρ1 u2ρ2

Consider flow in x-direction

Water with density ρ1 flows into the box

with velocity u1

Water with density ρ2 flows out of the box

with velocity u2

Volume flux into the box = [ u1 × Δy × Δz ]So mass flux = [ ρ1 × u1 × Δy × Δz ]

Volume flux out of the box = [ u2 × Δy × Δz ]So mass flux = [ ρ2 × u2 × Δy × Δz ]

6

ΔxΔy

Δzu1ρ1 u2ρ2

This can be written more generally as:

In the x-direction have:

ΔxΔy

Δz

v1ρ1

v2ρ2

The same in the y -direction

ΔxΔy

Δz

w1ρ1

w2ρ2

And in the z -direction

7

ΔxΔy

Δz

Putting this all together gives the continuity equation

Change in mass Convergence or divergence in flux

8

Remember from Calculus

This can be simplified

9

Eulerian Measurements

102810291030103110321033

In this Eulerian reference frame, density appears to be increasing

x

In fluid mechanics, measurements made in a

fixed reference frame to the flow are called Eulerian.

102810291030103110321033

x

10

Lagrangian Measurements

102810291030103110321033

In fluid mechanics, measurements made in a

reference frame that moves with the fluid are referred to as

Lagrangian.

102810291030103110321033

In this Lagrangian reference frame, density appears to be constant 11

In fluid mechanics, the reference frame is crucial to what is observed

In the previous example for the Eulerian reference frame, the scalar quantity in the fixed box is increasing because of advection.

How can this be represented mathematically?

102810291030103110321033

This is referred to as the local rate of change

u

x12

In a fixed or Eulerian reference frame, the local rate of change is influenced by advection in all three directions

In contrast, in a Lagrangian or a reference frame moving with the flow, there are no advective changes.

Local derivative no local sources or sinks

Eulerian Material derivative:

So, the total rate of change in a moving reference frame includes the local rate of change minus the advective contribution.

13

Previous derivation of continuity gives

Know that the Material derivative is related to the local derivative by:

This results in the continuity equation conservation mass

14

Conservation of volume

The Continuity Equation

One important characteristic of oceanic flows is that, even when density stratification is fundamental to the flow, density variations are still small, only a few

parts per thousand.

ρo ~ 1020 kg/m3

<ρ> ~ 2 kg/m3

ρ' < 2 kg/m3

In most situations, this term is way smaller than … this term

15

Introduction to Scaling

Scaling is a way to compare the relative importance of terms in an equation.

Estimate the relative importance of changes in density to convergences or divergences in the flow

Assume that changes in density can be estimated as:

The ratio of the density term to the convergence terms becomes:

And convergence can be estimated as:

= 2×10-3

16

Changes in density are negligible compared to convergences or divergences in the flow

17

This is the Boussinesq Approximation

It states that density differences are sufficiently small to be neglected, except where they appear in terms multiplied by g, the acceleration due to gravity.

It represented mathematically, as:

Assumes that the compressibility of seawater can be ignored in many situations.

Applying the Boussinesq Approximation gives the incompressible form of the continuity equation :

18

The Continuity Equation

A divergence in the flow in one direction, must be balanced by a convergence in the flow in another direction.

Consider a convergent flow over a fixed boundary:

No flow through boundary.19

Consider the box below Ignore flow in the y-direction:

The Continuity Equation

If there is convergent flow in the x-direction, what happens to the water surface?

20

z = -h

z = η

z = -h

z = η

Depth Integrated form of Continuity

To first approximation:

21

Wave propagation can be explained in terms of the depth averaged continuity equation:

convergencedivergence divergence

The water at any given point simply oscillates back and forth (no water is transported), but wave form propagates (energy is transmitted)

22

Upwelling and Downwelling

Upwelling is the upward motion of water caused by surface divergence. This motion brings cold, nutrient rich water towards the surface.

Downwelling is downward motion of water caused by surface convergence. It supplies the deeper ocean with dissolved gases.

Downwelling:Upwelling:

23

Next Class

• Conservation of tracers (heat and salt) – Chapter 4 – Knauss– Chapter 5 – Talley et al.

24