ATOC 4720 class35 1. The thermodynamic energy equation 2. The continuity equation

Previous classes Horizontal: vector form: Components: Vertical equation of motion:

Brief Review of previous a few classes: Geostrophic wind value: T- days Mid-latitude

Cross-isobar flow due to friction

The gradient wind

Smaller-scale convection: GEOSTROPHY BREAKS DOWN z y x Observations for small-scale convection: Velocity U-V: 20 m/s; Time T (100-1000s)= S

Hydrostatic balance Hydrostatic balance is well satisfied even by mesoscale convection.

Thermal wind relation

1. Thermodynamic energy equation Rewrite hydrostatic equation: (show math on blackboard)

Obviously, we have 3 equations, 4 unknowns We need an equation for T. [Prompt]

The first law of thermodynamics says: Denote as the heating rate:

Since So, Substituting into the above equation and denote Where,

Physics: Temperature change is determined by: [1] the rate of adiabatic heating or cooling due to compression or expansion; [2] the rate of diabetic heating.

Scale analysis: estimating the relative important of adiabatic and diabetic heating: [1]: Adiabatic heating: Typical pressure change over the course of a day following an air parcel In mid latitude mid-troposphere,

[2] Diabetic heating: absorption of solar radiation, absorption and emission of infrared radiation, latent heat release, in upper atmosphere,heat absorbed or liberated in chemical & photochemical reactions. Diabetic mixing with environment (latent & sensible). In lower atmosphere, sources and sinks tend to balance each other. As a result,

Note that all time-dependent term we introduced so far is: Time change following an individual air parcel. (Lagrangian) In most cases, we wish to know time change at a Specific location, say T change over Boulder. Local change: Eularian change.

Since We obtain Because, We have

3-d advection Local change Individual change Cold T Warm T Tb Ta Boulder Mountain in the west