Balanced Flow • The pressure and velocity distributions in atmospheric systems 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 Unit vector k ˆ points upward. ˆ ˆ t nˆ tˆ k nˆ 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 where 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 R radius of curvature (positive in = 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 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 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 of curvature R is also constant. 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. 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 V ∂Φ g fV = − g ∂n Φ0 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 approximately constant. Air parcels will follow circular paths in anticyclonic rotation with period 2πR 2π P = = V f 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 is imaginary ∂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. 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 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 balance of forces normal to flow. 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..