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Understanding the Continuity Equation in Atmospheric Dynamics

Learn about mass conservation in the atmosphere using different forms of the continuity equation: mass divergence form and velocity divergence form. Explore how to apply Euler's relation and the kinematic method to study vertical velocity. Discover the importance of compressibility in atmospheric dynamics.

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Understanding the Continuity Equation in Atmospheric Dynamics

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  1. Continuity Equation

  2. Statement of conservation of mass • Several different forms: Mass Divergence Form “Convergence of Density” Change of density with respect to time We can derive the “Velocity Divergence Form” by using the product rule and Euler’s Relation

  3. Move all terms to the left hand side • Apply product rule to mass flux (density * wind) terms • Apply Euler’s relation to simplify

  4. Velocity Divergence Form This works great for water bodies, since they are incompressible (density does not change). This makes the left hand side = 0 and the velocity divergence form becomes: But the atmosphere is compressible. So another approach is to use pressure (p) as the vertical coordinate, rather than height (z). Then the velocity divergence form becomes:

  5. Here, ω is the change in the vertical velocity with respect to pressure (whereas ‘w’ was the vertical velocity in height coordinates This equation can be used to compute the vertical velocity at various levels in the atmosphere. This technique is called the kinematic method because it only requires information about the winds. Integrate both sides from p1 to p2 Simplify Rearrange

  6. Vertical velocity at p2 Horizontal divergence between p1 and p2 Vertical velocity at p1 If the horizontal divergence is constant between p1 and p2, we can represent this equation in finite difference form:

  7. Lab Notes See revised question 1 on website For question 4,

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