Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove fluid – it would be… inflow rate – outflow rate

Now consider a small cube, fixed in space, through which air is flowing The x direction mass flux is given by

Using a Taylor series expansion about the center point, the rate of inflow through side A would be: and through side B would be: As with the basin, the rate of accumulation of mass in the cube must be:

Similarly: The net mass accumulation in the cube would be: Dividing by the volume

This equation can be written as: The flux form of the mass continuity equation Using the vector identity: this equation can be written as: or: The velocity divergence form of the mass continuity equation

The flux form of the mass continuity equation The velocity divergence form of the mass continuity equation For the special case of an incompressible fluid, density doesn’t change following parcel motion so the continuity equation reduces to: The incompressible form of the mass continuity equation Finally, let’s derive the mass continuity equation in pressure coordinates…

Consider a volume of air: Using the hydrostatic equation or We can write our volume as: and the mass of the parcel as: If we follow parcel motion, the mass of parcel should be conserved:

Divide both sides by Apply chain rule to RHS Recalling that The equation can be reduced, in the limit that the volume approaches 0, to be The pressure coordinate form of the mass continuity equation

The flux form of the mass continuity equation The velocity divergence form of the mass continuity equation The incompressible form of the mass continuity equation The various forms of the mass continuity equation so far

Finally, substituting the geostrophic and ageostrophic wind: The ageostrophic wind form of the mass continuity equation

The pressure coordinate form of the mass continuity equation The flux form of the mass continuity equation The velocity divergence form of the mass continuity equation The incompressible form of the mass continuity equation The various forms of the mass continuity equation The ageostrophic wind form of the mass continuity equation

The pressure coordinate form is particularly enlightening We can integrate this equation between the top of a column and the surface

If there is no vertical air motion at the surface and there is convergence, air will rise, a direct outcome of mass continuity. Vertical motion and column divergence patterns are directly related.

Conservation of energy We will consider mechanical energy, thermal energy, and total energy

The complete momentum equations Multiply (1) by u, (2) by v, and (3) by w to get energy equations

Add equations together: note that earth curvature and Coriolis force terms all cancel! Note that:Move this to left side

Kinetic Energy Potential Energy Work done by PFG Work done by Friction A change in total mechanical energy of a parcel of air must come about by work done by the pressure gradient and frictional forces Note that in geostrophic flow so the first term on the LHS is zero in geostrophic flow. Only the Ageostrophic wind component does work.

The thermodynamic energy equation First law of thermodynamics can be expressed as: or where is the diabatic heating rate and is the specific volume simplifying friction notation Mechanical energy equation Total energy equation (add TEE and MEE)

Note that:and So: Substituting: The Energy Equation

If flow is adiabatic, frictionless and steady state, then: -For an atmosphere at rest, an increase in elevation results in a decrease in hydrostatic pressure Special case of Bernoulli’s equation for incompressible flow

Other implications In accelerating flow over a hill the pressure difference between p 2 and p 1 must be > hydrostatic

Potential Temperature Temperature a parcel of air would have if it were brought dry adiabatically to a pressure of 1000 mb. “Dry adiabatically” implies No exchange of mass or energy with the environment, and no Condensation or evaporation occurring within the air parcel. Potential temperature equation derived from 1 st law of thermodynamics: S = Entropy c p = Specific Heat at constant pressure R d = dry air gas constant For an adiabatic process, dS = 0

Integrate equation from an arbitrary temperature and pressure to the potential temperature  and a pressure of 1000 mb.