Goodrich METR 324 Final

Lagrangian momentum

Change in momentum with time following a parcel

Eulerian momentum

Change in momentum with time at a fixed location.

Equations of motion

Derived from Newton’s second law, these equations describe the forces that cause the air in the atmosphere to move. They include advection, pressure gradient force, Coriolis force, and turbulent drag.

Advection

The movement of wind caused by other winds.

F_xAD = -u*(∆u/∆x) -v*(∆u/∆y)

F_yAD = -u*(∆v/∆x) -v*(∆v/∆y)

Pressure gradient force (PGF)

The only force that can initiate motion; it is perpendicular to isobars and causes air to flow from high pressure to low pressure.

F_xPG = -(1/𝜌)*(∆P/∆x)

F_yPG = -(1/𝜌)*(∆P/∆y)

Coriolis force

Force caused by the rotation of the Earth. (f_c = 2Ω * sin𝜙)

F_xCF = f_c*v

F_yCF = -f_c*u

Turbulent drag

Force that slows air down due to friction with the surface. Only present in boundary layer (bottom ~1km of atmosphere).

F_xTD = -w_T*(u/z_i). w_T = turbulent transport velocity

F_yTD = -w_T*(v/z_i) z_i = boundary layer depth

Steady state winds

Also known as balanced flow, they are winds that do not change with time (no acceleration).

(∆u/∆t) = 0 (∆v/∆t) = 0

Equations of motion sum to equal 0

Examples: geostrophic, ageostrophic, gradient

Geostrophic wind (u_g,v_g)

Steady state wind that blows parallel to isobars driven by a balance in PGF and CF.

v_g = (1/𝜌*f_c)*(∆P/∆x)

u_g = -(1/𝜌*f_c)*(∆P/∆y)

G =\sqrt{u_{g}^2+v_{g}^2}

Hydrostatic equation

Describes the relationship between pressure and depth in a static fluid. Can be used to find (u_g,v_g) on a constant pressure surface.

(∆P/∆z) = -𝜌g

Ageostrophic wind

The component of the actual wind that deviates from the geostrophic wind. Brings about the curvature of flow and jet streaks.

u_ag = u - u_g

v_ag = v - v_g

Gradient wind

Steady state wind around high or low pressure centers (ridges/troughs). Sub-geostrophic around lows, super-geostrophic around highs. Imbalance between PGF and CF to balance CN.

s = +1 for low pressure

s = -1 for high pressure

Rossby number

Number that relates inertial forces to the CF for atmospheric flow. As R_oc decreases, flow becomes geostrophic

R_oc = (G/f_c*R)

Continuity equation

Describes the conservation of mass in the atmosphere.

Thermal wind

Theoretical wind that blows parallel to isobars. Describes the vector difference between geostrophic winds at different pressure surfaces.

Relative vorticity

Measure of rotation of air about a vertical axis

Solid body rotation

Fluid rotates with constant angular velocity

Absolute vorticity

Relative vorticity + Coriolis

Potential vorticity

Absolute vorticity throughout the depth of a rotating column

Stretching term

Describes the deformation of a column of rotating air.