• The pressure and velocity distributions in atmospheric syyyyp,ppstems are related by relatively simple, approximate force balances. • We can gain a qualitative understanding by considering steady-state conditions, in which the fluid flow does not vary with time, and by assuming there are no vertical motions. • To explore these balanced flow conditions, it is useful to define a new coordinate system, known as natural coordinates.
Natural Coordinates
• Natural coordinates are defined by a set of mutually orthogonal unit vectors whose orientation depends on the direction of the flow.
Unit vector tˆ points along the direction of the flow. ˆ k kˆ Unit vector nˆ is perpendicular to ˆ t the flow, with positive to the left. kˆ nˆ ˆ t nˆ UiUnit vector k ˆ po ints upwar d. ˆ ˆ t nˆ tˆ k
nˆ
1 ˆ r k kˆ Horizontal velocity: V = Vtˆ tˆ kˆ nˆ ˆ V is the horizontal speed, t nˆ ˆ which is a nonnegative tˆ tˆ k nˆ scalar defined by V ≡ ds dt , nˆ whithhere s ( x , y , t ) is the curve followed by a fluid parcel moving in the horizontal plane. To determine acceleration following the fluid motion, r dV d = (Vˆt ) dt dt r dV dV dˆt = ˆt +V dt dt dt
δt δs δtˆ δψ = = = δtˆ t+δt R tˆ δψ δψ
radius of curvature (positive in R = R δs positive n direction) t n R > 0 if air parcels turn toward left R < 0 if air parcels turn toward right ˆ dtˆ nˆ k kˆ = (taking limit as δs → 0) tˆ R <0< 0 ds R kˆ nˆ ˆ ˆ ˆ t nˆ dt dt ds nˆ ˆ ˆ = = V t nˆ tˆ k dt ds dt R R > 0 nˆ
2 r dV dV dˆt = ˆt +V dt dt dt r 2 dV dV V vector form of acceleration following = ˆt + nˆ dt dt R fluid motion in natural coordinates
r − fkˆ ×V = − fVnˆ Coriolis (always acts normal to flow)
⎛ˆ ∂Φ ∂Φ ⎞ − ∇ pΦ = −⎜t + nˆ ⎟ pressure gradient ⎝ ∂s ∂n ⎠ dV ∂Φ = − dt ∂s component equations of horizontal V 2 ∂Φ momentum equation (isobaric) in + fV = − natural coordinate system R ∂n
dV ∂Φ = − Balance of forces parallel to flow. dt ∂s V 2 ∂Φ + fV = − Balance of forces normal to flow. R ∂n
∂Φ For motion parallel to geopotential height contours, = 0 , which means ∂s that the speed is constant following the motion.
If the geopotential gradient normal to the direction of motion is constant along a trajectory, the normal component equation implies that the radius oocuatuef curvature R is al so co nsta n t.
When these assumptions are met we can define several simple categories of balanced flow that depend on the relative contributions of the three terms in the normal component equation.
3 Geostrophic Flow
Straight-line flow parallel to the height contours ( R → ±∞ ).
0 V 2 ∂Φ + fV = − R ∂n
Horizontal components of Coriolis force and pressure gradient force are in exact balance. PGF
∂Φ Vg fV = − Φ g ∂n 0
Coriolis Φ0 + ΔΦ
Inertial Flow If the geopotential field is uniform on a constant pressure surface, then 0 V 2 ∂Φ V + fV = − R = − R ∂n f
Because uniform geopotential implies constant speed, then the radius of curvature is constant if we assume f is approximatel y constant. Air parcels will follow circular paths in anticyclonic rotation with period 2πR 2π P = = V f
4 Cyclostrophic Flow
If the horizontal scale of an atmospheric disturbance is sufficiently small, such as in a tornado, waterspout, or dust devil, then the Coriolis term will be substantially smaller than the pressure gradient and centrifugal terms:
2 0 1 2 V ∂Φ ⎛ ∂Φ ⎞ + fV = − V = ⎜− R ⎟ R ∂n ⎝ ∂n ⎠
There are four possible orientations for the direction of curvature and the pressure gradient: ∂Φ ∂Φ R > 0, < 0 V is real R < 0, < 0 V iiis imag inary ∂n ∂n ∂Φ ∂Φ R > 0, > 0 V is imaginary R < 0, > 0 V is real ∂n ∂n
V
PGF Centrifugal L
∂Φ Only low pressure R > 0, < 0 systems can have ∂n cyclostrophic flow. ∂Φ R < 0, > 0 ∂n
PGF Centrifugal L
Cyclostrophic flow can either be V clockwise or counterclockwise.
5 Cyclostrophic Flow in the Real World
• Because the Coriolis acceleration is neglected in cyclostrophic flow, there should be no preferred direction of rotation. • In fact, waterspouts and (especially) dust devils can show both clockwise and counterclockwise rotation. • Most (but not all) tornadoes in the Northern Hemisphere rotate counterclockwise, because they develop from large, rotating supercell thunderstorms. Because of their relatively large scale (~10 km), supercell thunderstorms do feel the effects of Coriolis acceleration.
Gradient Flow
For large-scale weather disturbances in which the flow is not required to be in a straight line, a three-way balance among the Coriolis, centrifugal, and pressure gradient forces exists.
1 2 V 2 ∂Φ fR ⎛ f 2 R 2 ∂Φ ⎞ + fV = − V = − ± ⎜ − R ⎟ R ∂n 2 ⎝ 4 ∂n ⎠
There are four possible orientations for the direction of curvature and the pressure gradient, each with two roots:
R > 0 R < 0 Positive root: unphysical Positive root: anomalous low ∂Φ ∂n > 0 Negative root: unphysical Negative root: unphysical Positive root: regular low Positive root: anomalous high ∂Φ ∂n < 0 Negative root: unphysical Negative root: regular high
6 regular V regular low high
PGF Centrifugal Centrifugal H L Coriolis Coriolis PGF
V ∂Φ ∂Φ R > 0, < 0, positive root R < 0, < 0, negative root ∂n ∂n anomalous anomalous low high
PGF Centrifugal Centrifugal L H Coriolis Coriolis PGF
V V ∂Φ ∂Φ R < 0, > 0, positive root R < 0, < 0, positive root ∂n ∂n
Geostrophic Wind vs. Gradient Wind
V 2 ∂Φ + fV = − Start with b al ance of f orces normal t o fl ow. R ∂n V 2 + fV − fV = 0 Rewrite using definition of geostrophic wind. R g V V g =1+ Solve for ratio of geostrophic wind to gradient wind. V fR
Rossby For normal cyclonic flow (fR > 0) the geostrophic wind overestimates number the gradient wind, while for anticyclonic flow (fR < 0) the geostrophic wind underestimates the gradient wind. The geostrophic wind is a good estimate of the gradient wind when the Rossby number is small.
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