Balance WindsGeostrophic BalanceCoriolis TermGradient WindCyclostrophic Wind BalanceBalance with FrictionSummary of Balance WindsContinuity EquationLevel of NondivergencePressure Tendency EquationBaroclinity (Baroclinicity) and BarotropyVorticityComponents of VorticityRelative and Absolute VorticityVorticity EquationDivergence TermTiltingBaroclinic TermRelative Vorticity in Natural CoordinatesSome Issues to Look AtContinuity EquationVertical Vorticity PlotsPrimitive Equations in Isobaric CoordinatesThe Omega EquationNotesFirst Term:Vorticity AdvectionSecond Term: Temperature AdvectionThird Term: Diabatic HeatingSummaryRules of Thumb for Synoptic AnalysesGeneral NotesThe Q VectorPetterssen’s Development EquationThe Thermal Wind EquationImplications of the Thermal Wind RelationsPotential VorticityRossby (Barotropic) Potential VorticityErtel’s Potential VorticityUses of PVMidterm Exam: General NotesChapter 2 OutlineChapter 3 OutlineChapter 4 OutlineBalance Winds• Geostrophic Balance• Gradient Wind• Cyclostrophic Wind Balance• Balance with Friction / Ekman BalanceGeostrophic BalanceThe equation of motion can be written as:)k-gG( vector nalgravitatio theis Gfriction of effects therepresents F termCoriolis thecalled is V2earth theof raterotation the toingcorrespond day/2pole)north thefrom upward pointing (positiveearth theoflocity angular ve theis whereGFV2p1dtVdrrrrrrrrrrrr=×Ωπ=ΩΩ+−×Ω−∇ρ−=Coriolis Term[]φΩ=φΩ=+−−=×Ω−∴φΩ+φΩ+φΩφΩ=φΩφΩ−=×Ω−ins2f and osc2fˆ wherekufˆjfui)wfˆfv(V2 ku)osc(-ju)ins(iv)sin-wosc(2- wvusincos0kji2V2rrrrrrrrrrlatitude theis and cos2fˆ and sin2f whereFvfuyp1dtdvFuwfˆfvxp1dtdu:as written be thereforecan motion ofequation theof components horizontal TheφφΩ=φΩ=−−∂∂ρ−=−−+∂∂ρ−=For the case of:• no friction (Fu = Fv ≡ 0)• no acceleration (du/dt = dv/dt ≡ 0)• |u|, |v| >> w which is typical on the synoptic scalethenvxpf1 v fvxp10uypf1-u fuyp10gg≡∂∂ρ=⇒+∂∂ρ−=≡∂∂ρ=⇒−∂∂ρ−=where ugand vgare the geostrophic wind components defined by these relations. The geostrophic wind relation can be written in vector notation as:)yjxi(operator gradient horizontal theis wherepf1kVzzg∂∂+∂∂∇∇ρ×=rrrr0ypf1xpf1100kjijviuV:product crossvector theof definition theusing checked becan Thispf1kVgggzg∂∂ρ∂∂ρ=+=∇ρ×=rrrrrpf1kVzg∇ρ×=rr windcgeostrophi for thestraight is flow that theassumed isIt latitudes lowat achievedrarely is balance cGeostrophi latitudes lowat 0 f because latitudeslower at Vstronger in result willp of egiven valuA )Vlarger theplarger (thegradient pressure the toalproportionlinearlyalmost is V height,given aat constant nearly is As hemispheresouthern in theleft the toand hemispherenorthern in theright the tois pressurehigh thesuch that directed is V isobars) the toparallel isdirection wind (the p lar toperpendicu and horizontal bemust V force Coriolis theand forcegradient pressureebetween thbalancetheis windcgeostrophi The gzgzggzg••→∇•∇ρ••∇••rrrrrxpf1vypf1-ugg∂∂ρ=∂∂ρ=Gradient Wind• The geostrophic wind balance assumes that the wind flow is straight• A more general form of a balanced wind can be obtained if accelerations due to curvature in the height or pressure fields are taken into account (remember that changes in the direction of the velocity vector with time result in acceleration)• The gradient wind like the geostrophic wind is frictionless, but it is not unacceleratedforce lcentrifuga simply the isequation above theof side handright Ther of magnitude theis R ectorposition v theis r indgradient wfor standsgr subscript thewhere)(dtVd:curvature todueon accelerati for the for the expressionan obtain can wenotes sPielke' Following wind.cgeostrophi for the it was as zeronot is dtVd termthecurvature, of effects the todueon accelerati have now weAs21dtVd:aswritten becan motion ofequation TheT2rrrrrrrrrrrrRVGFVpTgr−=+−×Ω−∇−=ρThe direction of the unit vector at points A, B, C and D are denoted by the appropriate Cartesian unit vector)r(Rvujdtdvidtdu )r(RVdtVdT2gr2grT2grrrrrrr−+=+⇒−=rr−T2grRudtdv0dtdu−==T2grRvdtdu0dtdv−==T2grRudtdv0dtdu==T2grRvdtdu0dtdv==• To investigate the balanced wind which develops when the acceleration due to curvature is included along with the Coriolis force and the pressure gradient force, we will focus on point A.There is no loss of generality as the coordinate system can always be rotated so that a point of interest corresponds to location A. • At point A, ug= 0, while vg> 0 for a low and vg< 0 for a high pressure in the northern hemisphere. Substituting into:and ignoring friction and the fw Coriolis term we get:fvxpdtdu+∂∂−=ρ1grT2grfvxp1Rv+∂∂ρ−=−)(:get we1- as 122ggrgrgTgrggrTgrvvffvfvRvfvxpNowfvxpRv−=+−=−−=∂∂+∂∂−=−ρρ• The velocity vgrwhich solves this relation is called the GRADIENT WINDgrgTgrfvfvRv+−=−2• The gradient wind balance is a three-way balance between the Coriolis force, the centrifugal force and the horizontal pressure gradient forceCentrifugal forcePressure gradient forceCoriolis force2/)4(:for v solve toformulaequation quadratic theUsing0v:equation thisgRearrangin)(22gr2gr2gTTTgrgTgrTggrgrgTgrfvRRffRvfvRvfRvvffvfvRv+±−==−+−=+−=−• For a cyclone in the northern hemisphere, vg> 0 at A so that the radical is always real => no limit to the magnitude of the gradient wind• For an anticyclone in the northern hemisphere, vg< 0 at A so that f2RT2> 4RTfvg => vg< fRT/4 for the radical to be real => there is a constraint on the magnitude of the pressure gradient force in anticyclones that does not exist for low pressures. This is the reason that lows on synoptic weather maps often have tight gradients while highs don’t.ggrT2grgTgrT2grvvfRv:asrewritten becan 0fvRvfRv=+=−+• vgr< vgfor a cyclone as vg> 0 at A •|vgr| > |vg| for an anticyclone as vg< 0 at A• These inequalities show that for the same pressure gradient (as represented by the geostrophic wind), the gradient balanced wind is stronger around a high than a low• The gradient winds associated with a cyclone are SUBGEOSTROPHICbecause the centrifugal force helps to balance the acceleration due to the pressure gradient force, therefore the Coriolis terms fvgrand vgrdon’t need