Conservation of Mass – The Continuity Equation

1. Conservation of Mass – The Continuity Equation Lecture 6 OEAS-604 October 3, 2011 • Outline: • Conservation of Mass – Flux in = Flux out • Cartesian Coordinate System • Derivation of Continuity Equation • Eulerian vs. Langrangian Reference Frames • The Boussinesq Approximation • Incompressible form of the Continuity • Application of Continuity

2. Conservation of Mass change in mass = flux in – flux out mass = density × volume change in mass = flux in • 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? • How fast is volume changing? • How fast is water rising? Δz Δy Δx flux in = flux out

3. Cartesian Coordinate System z y The location of any point in space can be uniquely described by its coordinates (x,y,z) x Similarly the velocity vector of an object can be uniquely described in Cartesian Coordinates Typically we use “u” to denote a vector in the x-direction “v” to denote a vector in the y-direction, and “w” to denote a vector in the z-direction.

4. Conservation of Mass in a Cartesian Coordinate System ρ Δz Δy Δx If box is filled with water of density ρ, what is the mass of water in the box? Mass = ρ × Δx × Δy × Δz

5. Conservation of Mass change in mass = flux in – flux out Considering only flow in x-direction for now! Water with density ρ1 is flowing into the box with velocity u1 Water with density ρ2 is flowing out of the box with velocity u2 u1ρ1 u2ρ2 Δz Δy Δx Volume flux of water into the box = [ u1 × Δy × Δz ]; So mass flux = [ ρ1 × u1 × Δy × Δz ] Volume flux of water out of the box = [ u2 × Δy × Δz ]; So mass flux = [ ρ2 × u2 × Δy × Δz ]

6. In the x-direction we have: u1ρ1 u2ρ2 Δz Δy This can be written more generally as: Δx v2ρ2 The same holds true in the y -direction giving: Δz v1ρ1 Δy Δx w2ρ2 And in the z -direction giving: Δz Δy Δx w1ρ1

7. Putting this all together gives the continuity equation: Convergence or divergence in flux Change in mass Δz Δy Δx

8. This can be simplified further!! Remember from Calculus that

9. Eulerian Measurements 1033 1032 1031 1030 1029 1028 x In fluid mechanics, measurements made in a fixed reference frame to the flow are called Eulerian. 1033 1032 1031 1030 1029 1028 x In this eulerian reference frame, density appears to be increasing

10. Lagrangian Measurements 1033 1032 1031 1030 1029 1028 In fluid mechanics, measurements made in a reference frame that moves with the fluid are referred to as Lagrangian. 1033 1032 1031 1030 1029 1028 In this lagrangian reference frame, density appears to be constant

11. In fluid mechanics, your reference frame is crucial to what you observe 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? This is referred to as the local rate of change u 1033 1032 1031 1030 1029 1028 x

12. In a fixed or Eulerian reference frame, the local rate of change is influenced by advection in all three directions Local derivative: In contrast, in a Lagrangian or a reference frame moving with the flow, there are no advective changes. Material derivative: So, the total rate of change in a moving reference frame includes the local rate of change minus the advective contribution.

13. From our previous derivation of continuity we have: We also, know that the Material derivative is related to the local derivative by: This results in the “continuity equation”

14. The Continuity Equation In most situations, this term is way smaller than … this term 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

15. Quick Introduction to Scaling: Scaling is a very back of envelope way of comparing the relative importance of terms in an equation. In this case, we are interested in estimating the relative importance of changes in density to convergences or divergences in the flow: If we assume that changes in density can be estimated as: And convergence can be estimated as: The ratio of the density term to the convergence terms becomes: = 2×10-3!!

16. Bottom Line is that in terms of the continuity equation, changes in density are negligible compared to convergences or divergences in the flow!

17. This is called 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 also can be represented mathematically, as: This says that the compressibility of seawater can be ignored in many situations. So, 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. The Continuity Equation Consider the box below, and ignore flow in the y-direction: If there is convergent flow in the x-direction, what happens to the water surface?

20. Depth Integrated form of Continuity: To first approximation: z = η z = η z = -h z = -h

21. Wave propagation can be explained in terms of the depth averaged continuity equation: convergence divergence 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. Upwelling: Downwelling: