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Balanced Flows SO 254 β Spring 2017 LCDR Matt Burich Balanced Flows Before discussing states of balance between the various forces in the equation of motion, weβll first define a slightly different coordinate system called βnatural coordinatesβ which will maintain a consistent orientation relative to a moving air parcel Natural coordinates consist of unit vector components: π§ π§ π§: Normal to horizontal velocity and π oriented positive leftward π π: Tangential to horizontal velocity π€: Vertical unimportant for this discussion since weβll only be considering horizontal flows Balanced Flows In this system, if a distance traveled along the path of the parcel is πΏπ, then the horizontal velocity may be expressed as: π·π = ππ π·π‘ π·π 1 ππ and = π·π‘ π ππ π§ π π πΏπ Pressure gradient force along the motion path (Coriolis and centrifugal force act in the π§-direction so they do not exist here For our purposes, weβll assume the flow is unaccelerated thus there is no pressure gradient in the π-direction In the π§-direction we find the forces: π2 1 ππ + + ππ = 0 π π ππ centrifugal force pressure gradient force Coriolis π§ ππ ππ π·π π·π‘ = 0 and =0 Coriolis π PGF π : radius of curvature π β‘ 2Ξ© sin π centrf. Balanced Flows π2 1 ππ + + ππ = 0 π π ππ centrifugal force pressure gradient force Coriolis There are four possible balances that may exist between the forcesβ¦ three of which weβll consider directly 1) For flows with no curvature (π β β), centrifugal force is zero and we have: 1 ππ + ππ = 0 π ππ β 1 ππ = βππ π ππ L PGF Coriolis This balance of pressure gradient force and Coriolis is referred to as geostrophic balance height contours H 1 ππ = βππ π ππ PGF Geostrophic Balance (500 mb surface) Coriolis Low pressure Why does an air parcel begin to move? 528 dm An external force (PGF) is applied to it Once the parcel is moving, what additional force is acting? PGF velocity Coriolis Forces in balance North 540 dm 546 dm 552 dm 558 dm Coriolisβ¦to the right of direction of motion (Northern Hemisphere) PGF never changes in this example (equally spaced height contours) but the magnitude of Coriolis is dependent upon parcel velocity 534 dm High pressure PGF 500 mb North (side view) 558 dm 552 dm 546 dm 540 dm 534 dm 528 dm 1 ππ = βππ π ππ PGF Geostrophic Balance Coriolis In reality, height contours are rarely perfectly straight, so the state of geostrophic balance is an approximation The validity of the βgeostrophic approximationβ for a particular flow situation is checked via computation of the Rossby π number: π 0 = ππΏ π = characteristic velocity scale πΏ = characteristic length scale π = 2Ξ© sin π The smaller the value of π 0 , the more valid the geostrophic approximation For large scale motions: π~10 m s β1 , πΏ~106 m, and π~10β4 π β1 (in midlatitudes) Thus, π 0 ~0.1 (a reasonable approximation) Balanced Flows π2 1 ππ + + ππ = 0 π π ππ centrifugal force 2) pressure gradient force Coriolis For flows where Coriolis force is insignificant, we have: π2 1 ππ + = 0 π π ππ β 1 ππ π2 = β π ππ π These are generally small scale flows Compute the Rossby number for a tornado: π~100 m s β1 , πΏ~102 m, and π~10β4 π β1 π 0 ~104 L PGF Centrf. Not geostrophic! height contours H Balanced Flows π2 1 ππ + + ππ = 0 π π ππ centrifugal force 2) pressure gradient force Coriolis For flows where Coriolis force is insignificant, we have: π2 1 ππ + = 0 π π ππ β 1 ππ π2 = β π ππ π These are generally small scale flows This balance of pressure gradient force and centrifugal force is referred to as cyclostrophic balance L PGF Centrf. height contours H Balanced Flows π2 1 ππ + + ππ = 0 π π ππ centrifugal force 3) pressure gradient force Coriolis For flows where all three forces are significant, we have: 1 ππ π2 = β β ππ π ππ π These are often medium (meso) scale flows Compute the Rossby number for a hurricane: π~50 m s β1 , πΏ~105 m, and π~10β4 π β1 L Centrf. PGF Coriolis π 0 ~5 Neither very large nor very small height contours H Balanced Flows π2 1 ππ + + ππ = 0 π π ππ centrifugal force 3) pressure gradient force Coriolis For flows where all three forces are significant, we have: 1 ππ π2 = β β ππ π ππ π These are often medium (meso) scale flows This balance of pressure gradient force, Coriolis, and centrifugal force is referred to as gradient (wind) balance L Centrf. PGF Coriolis height contours H Summary π2 1 ππ + + ππ = 0 π ππ π centrifugal force pressure gradient force Coriolis Small scale rotation (e.g., tornado / dust devil) Geostrophic balance Gradient (wind) balance Cyclostrophic balance 1 ππ = βππ π ππ 1 ππ π2 = β β ππ For π ππexample, π jet stream flow that is relatively straight π 0 βͺ 1 π 0 β 1 π 0 β« 1 Large scale flows (synoptic scale) Medium scale flows (mesoscale) Small scale flows 1 ππ π2 = βjet For example, π ππ π stream flow that is strongly curved (at the base of a trough) What happens when friction gets involved? H L