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CLIMATE/EARTH 401�Geophysical Fluid Dynamics�Dynamical Meteorology

Richard B. Rood

rbrood@umich.edu

https://clasp.engin.umich.edu/people/richard-b-rood/

https://openclimate.org/

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Mass Conservation: Continuity

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Conservation of Mass

Conservation of mass leads the continuity equation
Continuity 🡪 Continuous
No holes in a fluid
This equation describes the time rate of change of mass (density)

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Remember our particle of atmosphere, our parcel

Δx

Δy

Δz

ρ ≡ density =

mass per unit volume (ΔV)

ΔV = ΔxΔyΔz

m = ρΔxΔyΔz

-------------------------------------

p ≡ pressure =

force per unit area

acting on the particle of

atmosphere

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The Eulerian point of view our parcel is a fixed volume and the fluid flows through it.

Δx

Δy

Δz

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Introduce mass flux, ρu

ρu = mass flux at (x, y, z) in the x direction.

Flux is mass per unit time per area
Mass flux out = �

.

Δx

Δy

Δz

(x, y, z)

x

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Introduce mass flux, ρu

ρu = mass flux at (x, y, z) in the x direction.

Flux is mass per unit time per area
Mass flux in = �

.

Δx

Δy

Δz

(x, y, z)

x

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What is the change of mass inside the fixed volume?

The change of mass in the box is equal to the mass that flows into the box minus the mass that flows out of the box = (flux) x (area)

Mass out right (downstream) face

Mass in left (upstream) face

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Extend to 3-Dimensions

The change of mass in the box is equal to the mass that flows into the box minus the mass that flows out of the box = (flux) x (area)

Note: this is change in mass per unit volume.

Recognizing the definition of divergence

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Eulerian Form of the Continuity Equation

Δx

Δy

Δz

In the Eulerian point of view, our parcel is a fixed volume and the fluid flows through it.

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The Lagrangian point of view is that the parcel is moving.

Δx

Δy

Δz

And it changes shape…

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Homework Exercise: Derive the Lagrangian Form

Remember, we can write the continuity equation

�

Use the chain rule (e.g., ) �to go from the above �equation to

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Lagrangian Form of the Continuity Equation

The change in mass (density) following the motion is equal to the divergence

Convergence = increase in density (compression)

Divergence = decrease in density (expansion)

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