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Mass Conservation and Continuity in Geophysical Fluid Dynamics

Learn about the conservation of mass and continuity equations in atmospheric dynamics, including the relationship between mass, velocity, and thermodynamic variables.

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Mass Conservation and Continuity in Geophysical Fluid Dynamics

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

  2. Weather • National Weather Service • Model forecasts: • Weather Underground • Model forecasts: • NCAR Research Applications Program

  3. Outline • Conservation of Mass • Continuity

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

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

  6. 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)

  7. 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

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

  9. (x, y, z) Dz . Dy Dx x 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 =

  10. 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

  11. Mass out right (downstream) face Mass in left (upstream) face 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)

  12. Note: this is change in mass per unit volume. Recognizing the definition of divergence 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)

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

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

  15. 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

  16. The change in mass (density) following the motion is equal to the divergence Convergence = increase in density (compression) Divergence = decrease in density (expansion) Lagrangian Form of the Continuity Equation

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

  18. 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…

  19. 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.

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