Suggested reading: Lackmann (2011), Sections

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1 QG Thery and Applicatins: Apprximatins and Equatins Atms 5110 Synptic Dynamic Meterlgy I Instructr: Jim Steenburgh jim.steenburgh@utah.edu Suite 480/Office 488 INSCC Suggested reading: Lackmann (2011), Sectins Mtivatin The desire fr a system f equatins that are simplified but retain the key dynamical prcesses necessary t describe, diagnse, and understand the behavir f largescale weather systems. Wrds f wisdm The equatins and their derivatin is admittedly difficult. It is imprtant that ne nt merely memrize the equatins by rte. It is much mre imprtant t understand what they mean physically. Hwie Bluestein Overarching assumptin f QG thery The Rssby Number (R0=U/fL) is small, which enables us t neglect the agestrphic wind in sme (but nt all) terms f the gverning equatins. Gverning equatins and their derivatin QG Thery is based n simplified versins f the equatins f mtin, cntinuity equatin, and the thermdynamic energy equatin. 1

2 Hrizntal mmentum equatin Derivatin 1. Break dwn the mmentum int gestrphic and agestrphic cmpnents V" = V" % + V" '% 2. Neglect the fllwing (by scale analysis) Frictin Advectin by the agestrphic wind and vertical velcity Advectin f agestrphic mmentum Lcal agestrphic mmentum tendency DV" Dt = V " t + V " V" = V " % t + V " '% t + V " % V" % + V" % V" '% + V" '% V" % + V" '% V" '% V" '% t, V " % V" '%, V" '% V" %, V" '% V" '% = 0 and in V" % V" % ω u % p, ω v % DV " Dt V " % t + V " % 5 V" % = DV " % where D = t + u % x + v % y 2

3 Physical interpretatin: The rate f change f mmentum fllwing parcel mtin is apprximately equal t the rate f change f gestrphic mmentum fllwing gestrphic mtin 3. Assume a midlatitude β-plane (Taylr expansin apprximatin fr f) f = f 9 + f y y + : f y : y : 2 + Ignring higher rder terms f = f 9 + f y y = f 9 + βy This allws f 9 t replace f in the gestrphic wind relatinship, which becmes V" % = 1 f 9 k@ Φ Or alternatively, f 9 k@ V" % = Φ 4. Use the abve assumptins and relatinships t rewrite the hrizntal mmentum equatin DV" Dt = Φ V" as DV" % = f 9 k@ V" % (f 9 + βy)k@ (V" % + V" '% ) = f 9 k@ V" % f 9 k@ V" % f 9 k@ V" '% βyk@ V" % βyk@ V" '% = f 9 k@ V" '% βyk@ V" % βyk@ V" '% 3

4 5. Using scale analysis it can be shwn that V" '% can be neglected, yielding the final frm f the QG mmentum equatin DV" % = f 9 k@ V" '% βyk@ V" % Physical interpretatin: The first right hand term represents the Crilis frce acting n the agestrphic wind, which in turn leads t an acceleratin f the gestrphic flw perpendicular t the agestrphic wind. I ve yet t cme up with a gd interpretatin f the secnd term n the right hand side! Hydrstatic Apprximatin (pressure crdinate frm) p = RT p Cntinuity Equatin Which can be written Since Prf: u x + v y + ω u '% x + v '% y + ω u % x + v % y = 0 u % = 1 f 9 y, V" % = 1 f 9 k@ Φ v % = 1 f 9 x u % x + v % y = x I 1 f 9 y J + y I 1 f 9 x J = 0 4

5 Physical interpretatin: The gestrphic wind is nndivergent. Divergence and vertical velcity are assciated with the agestrphic flw. Thermdynamic Energy Equatin Similar t mmentum, neglects the advectin f temperature by the agestrphic wind, hwever, the effects f vertical velcity are retained since vertical velcity has a majr influence n temperature thrugh adiabatic warming and cling. DT σp R ω = J C N where σ = RT dlnθ p dp = RT p 1 dθ θ dp Thus, σ is prprtinal t static stability. Typically we assume the flw is adiabatic (i.e., J=0), but it is pssible t include diabatic effects if desired (we wn t fr nw). Summary The QG mmentum equatin, gestrphic wind relatinship, hydrstatic apprximatin, cntinuity equatin, and thermdynamic energy equatin frm a clsed set f equatins fr the dependent variables Φ, V" %, V" '%, ω, and T. DV" % = f 9 k@ V" '% βyk@ V" % V" % = 1 f 9 k@ Φ p = RT p u '% x + v '% y + ω DT σp R ω = J C N 5

6 Gestrphic Relative Vrticity Derived by taking the curl f the gestrphic wind ζ % = V" % = 1 f 9 k@ Φ = 1 f 9 : Φ = g 9 f 9 : Z Key pints: Whenever yu see V W X : Φ, think ζ % Cyclnic (psitive in NH) ζ % assciated with minima in Z Anticyclnic (negative in NH) ζ % assciated with maxima in Z QG Vrticity Equatin Derived by taking the curl f the QG mmentum equatin (i.e., / x f the u % / t % equatin and / y f the v % / t % equatin) ζ % t = V ω " % Zζ % + f[ + f 9 p What changes the gestrphic relative vrticity at a pint? Gestrphic abslute vrticity advectin > 0 if advectin is cyclnic (i.e., psitive in NH) < 0 if advectin is anticyclnic (i.e., negative in NH) Stretching & cmpressin (r cnvergence and divergence) > 0 if stretching (NH) < 0 if cmpressin (NH) Alternatively, we can write the QG vrticity equatin as: Dζ % Dt = f ω 9 p 6

7 Which basically says that fllwing the flw, nly stretching (i.e., cnvergence) and cmpressin (i.e., divergence) cause the gestrphic abslute vrticity t increase r decrease, respectively. 7