Dynamics Exam 2

Rossby number

When Ro << 1, the Coriolis force is very important

time scales: τEarth′s rotation/τmotion

geostrophic balance

when there's no friction, non-accelerating flow (d~u/dt = 0) implies balance between coriolis and PGF

Ro << 1 (large-scale synoptic flow)

geostrophic wind in x and y coordinates

blows parallel to isobars

balanced circular flow

-circular flows in horizontal plane, using radial coordinate system, are subject to PGF, COR (from Earth's rotation), CF (from rotating flow)

-v is azimuthal (swirling/purely rotational) wind speed

- v>0 = counterclockwise

- 4 balances depending on strength of forces

gradient wind balance

Ro ∼ 1

balance bw PGF, COR, CF

other implications of gradient wind balance

-for same pressure and radius, gradient high has stronger winds

-for same wind speed, gradient low is deeper (larger pressure perturbations)

-gradient lows can (in theory) have counter clockwise or clockwise flow

-gradient highs can only have clockwise flow

(circular) geostrophic balance

Ro << 1

balance bw PGF and COR

eqn = 0

cyclostrophic balance

Ro >> 1

balance bw PGF and CF

only supports low pressure systems (CCW or CW flow)

inertial balance

balance between COR and CF

PGF small or 0

subgeostrophic

flow around a (regular) gradient low

supergeostrophic

flow around a gradient high

coriolis direction

right of motion in NH

PGF direction

always high to low pressure

pressure gradient is opposite

centrifugal force direction

ALWAYS outward

circular balances force diagram

gradient balance-

high: hypothetical coriolis in geostrophic = PGF, fv > fvg, v > vg (supergeostrophic)

low : coriolis is weaker than PGF, fv < fvg, v< vg (subgeostrophic)

cyclostrophic balance-

high : not possible bc PGF and CF point outward

low : possible ccw and cw flows

inertial balance-

direction on flow dependent on hemisphere

NH : f > 0, Vin < 0, clockwise

SH : f < 0, Vin > 0, counterclockwise

solutions to gradient wind balance

positve root: regular soln

negative root: anomalous soln

gradient wind balance 4 ways

**using solution to gradient wind balance equation

regular low:

+ sq root → v > 0, counter clockwise

anomalous low:

- sq root → v < 0, clockwise

regular high:

+ sq root → v > 0, clockwise

anomalous high

- sq root → v < 0, clockwise

cannot be counterclockwise because coriolis must be directed inward (3 forces cannot be directed outwards)

cyclonic highs are not possible

surface friction

modulates balanced flows by reducing wind speed and turning flow to be cross-isobaric (towards lows, away from highs)

vorticity

a microscopic measure describing the spin of fluid particles

circulation

- macroscopic measure of the rotation of a flow field

- S is a closed path

- dl is a distance increment tangent to the path and CCW

Stokes Theorem

implies that circulation is equivalent to the area integral of the VORTICITY perpendicular to the surface enclosed by path S

when dl = Rdθ and in the same direction as wind, C = the integral of 0 to 2π of |u|→Rdθ. equal to average vorticity within the circle times πR^2

Bjerknes Circulation Theorem

governs evolution of circulation of the flow relative to Earth's rotation

AE = Asinφ, projection of the area enclosed by the material circuit onto the equatorial plane

(Bjerknes) coriolis term

generates circulation through

1) changes in A bc of divergence/convergence of flow

2) changes in latitude (therefore projection of area) by changing orientation of circuit relative to Ω→(via meridional motion).

(Bjerknes) Pressure gradient term

aka barolinic and solenoidal term

-generates circulation if pfg is non uniform

-doesn't generate circulation if atmo is barotropic

-generates circulation in baroclinic fluid

barotropic state

- density is a function of pressure only, ρ = ρ(p)

- parallel density and pressure contours

- ∇ρ × ∇p = 0

- Pressure gradient term is 0

- (density or temperature doesn't vary on pressure surface) no temperature gradient , thus no vertical wind shear ∂u→g/∂p=0

- no thermal wind

baroclinic state

- density is not a function of pressure only, ρ = ρ(p, x, y, t)

- ∇p is not parallel to ∇ρ (or ∇T)

- ∇ρ × ∇p ≠ 0

- pressure gradient term is non-zero

- (density and temperature vary on pressure surface) there is a temp gradient, thus there is vertical shear ∂u→g/∂p ≠ 0

- there is thermal wind

equivalent barotropic state

- temp contours constant along height surface but vary long p surface

- speed of u→g varies with height, not direction

*technically baroclinic