Richard B. Rood (Room 2525, SRB) University of Michigan AOSS 401 Geophysical Fluid Dynamics: Atmospheric Dynamics Prepared: 20130919 Mass Conservation / Continuity Richard B. Rood (Room 2525, SRB) University of Michigan rbrood@umich.edu

Weather National Weather Service Weather Underground Model forecasts: Weather Underground NCAR Research Applications Program

Outline Conservation of Mass Continuity

Newton’s Law of Motion Where i represents the different types of forces.

Momentum Equations We have u, v, w, ρ, p which depend on (x, y, z, t).

Conservation of Mass Conservation of mass leads to another equation; the continuity equation Continuity  Continuous No holes in a fluid Another fundamental property of the atmosphere Need an equation that describes the time rate of change of mass (density)

Remember our particle of atmosphere, our parcel r ≡ density = mass per unit volume (DV) DV = DxDyDz m = rDxDyDz ------------------------------------- p ≡ pressure = force per unit area acting on the particle of atmosphere Dz Dy Dx

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

. Introduce mass flux, ρu (x, y, z) Dx Dy Dz (x, y, z) x ρu = mass flux at (x, y, z) in the x direction. Flux is mass per unit time per area Mass flux out =

. Introduce mass flux, ρu (x, y, z) ρu = mass flux at (x, y, z) in the x direction. Flux is mass per unit time per area Mass flux in = Dz . Dy Dx x

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

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

Eulerian Form of the Continuity Equation Dx Dy Dz In the Eulerian point of view, our parcel is a fixed volume and the fluid flows through it.

The Lagrangian point of view is that the parcel is moving. Dx Dy Dz And it changes shape…

In-Class 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

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)

The Lagrangian point of view is that the parcel is moving. Dx Dy Dz And it changes shape…

Our System of Equations We have u, v, w, ρ, p which depend on (x, y, z, t). We need one more equation for the time rate of change of pressure…

Summary The conservation of mass is one of three basic conservation laws we use in atmospheric dynamics Momentum Mass Energy The mass continuity equation connects mass, r, to the velocity field. Also connects the thermodynamic variables to the velocity (momentum) field.