Note: U is the 3D velocity The field is conserved following the - PowerPoint PPT Presentation

rate of change of a field following with the air parcel Rate of change associated with the movement of the fluid through the background temperature field rate of change at a fixed point on Earth Note: U is the 3D velocity

The field is conserved following the motion (that is each individual parcel retains conserves its temperature) and the local rate of change is entirely due to advection.

rate of change at a fixed point on Earth Rate of change associated with the movement of the fluid through the background temperature field rate of change of a field following the air parcel

Note: V = horizontal velocity Euler Lagrange to Euler: Hydrostatic Equilibrium 10 x larger Except in lower Boundary where all terms are small

In atmospheric sciences: the five fundamental equations that describe the evolution of large scale atmospheric motions.

Could be altitude, Z, potential temperature, θ , Pressure, P, or geopotential, Φ The geopotential is the gravitational potential per unit mass. Work done against planet’s gravity to raise a unit mass from the surface to a specific altitude above. Include effects of fall off of gravity with height, also centrifugal acceleration

The acceleration g g is normal to a surface of constant geopotential, an oblate spheroid, which outlines Earth, and compared to a sphere.

Total force per unit mass in the vertical direction = Pressure force + gravity + Coriolis “Force” + friction

Hydrostatic Equilibrium or, in pressure coordinates, by the hypsometric equation. The geopotential is the gravitational potential per unit mass. Work done against planet’s gravity to raise a unit mass from the surface to a specific altitude above.

“J dt” = rate of First law of thermodynamics diabatic heating α =1/V Add Equation of State: adiabatic expansion or compression effects of diabatic heat sources and sinks, e.g. effects of solar radiation, and absorption/emission of IR radiation

Integrate from the top of the atmosphere, P* to a pressure below Note: at the top of the atmosphere P=P* = 0 and ω = 0 (insignificant escape) Consider the Lagrangian expression for vertical motion: Let P=Ps and we have an expression for the surface boundary condition

Centifugal “force” Objects like to follow a straight path Coriolis “force” an object travelling away from the equator will be heading east faster than the ground. Objects travelling towards the equator will be going more slowly than the ground beneath them and will seem to be forced west. We will do this properly in next class

Prescribed Prognostic Prognostic Prognostic

Which has the smallest and largest Re:

Pr of air from 175-1900 K is 0.68-0.74

http://www.claymath.org/millennium-problems/navier–stokes-equation

Those are…