ζ a = V ζ a s ζ a φ p = ω p V h T = p R θ c p Derivation of the Quasigeostrophic Height Tendency and Omega Equations

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1 Derivatin f the Quasigestrphic Height Tendency and Omega Equatins Equatins Already Derived (x, y, p versins) Equatin f Cntinuity (Dines Cmpensatin): = ω Hypsmetric Equatin: T = p R φ Vrticity Equatin (natural crdinates): ζ a = V ζ a s ζ a Temperature Tendency Equatin T = ut x ω θ T = ut x ω p R θ c p T dq dt 1 c p dq dt The simplified vrticity equatin is ζ a = V ζ a s ζ a

2 (1) The cntinuity equatin in x,y,p is = ω (2) Nw, t cnvert (1) t a quasi-gestrphic vrticity equatin, we will make sne synptic-scaling arguments. First, we will replace the real wind with the gestrphic wind. Then, we will remember that the abslute gestrphic vrticity at a pint is dminated by the Crilis parameter, because relative GEOSTROPHIC vrticity values tend t range frm 3 t 3 X 10-5 s -1 while the Crilis parameter, f, is f rder10-4 s -1. Hence, the abslute gestrphic vrticity in the stretching (r divergence) term can be apprximated by the Crilis parameter, which, fr this simplificatin, is given the symbl f. S, substituting (2) int (1) and perfrming the quasi-gestrphic simplificatins yields ζ a g = V g ζ a g s f ω (3) Nw, recall frm Metr 430 s Lab #5 that (4a) ζ g = 1 f 2 Φ = g f 2 z ζ a g = 1 f 2 Φ f = g f 2 z f (4b) where ø =-gz, the geptential height. 2

3 Substituting (4a, b) int the left side f (3), remembering that the lcal tendency f f = 0 and expanding the abslute vrticity in the advectin term, yields ( 2 Φ) (5) = V g (ζ g f ) f 2 ω Quasigestrphic Vrticity Equatin where the vectr ntatin is used fr the hrizntal advectin term. This is the quasigestrphic vrticity equatin. It is valid t the extent that the wind is gestrphic and that the simplificatins made t the vrticity equatin based upn synptic scaling apply. The thermdynamic energy equatin is T = ut x ω θ θ c p T dq dt T = ut x ω p (6a,b) R 1 dq c p dt We can substitute the equatin f state int the hydrstatic law t btain the Hypsmetric Relatin (in x, y, p crdinates) T = p R φ (7) and place (7) int (6). Equatin (7) states that the temperature is prprtinal t the mean thickness f the layer centering n the level in quesitn. We can als assume that diabatic effects are small, substitute in the definitin f the static stability parameter, and assume that hat the wind is gestrphic t btain the quasigestrphic thermdynamic energy equatin,. 3

4 p φ R = V g ω p R (8) Quasigestrphic thermdynamic energy equatin We can simplify the left hand side f (8) by remembering that φ φ = φ = χ (9a,b) where X is the geptential height tendency (X<0 = height falls etc.). Equatins (9a,b) can be substituted int equatins (5) and (8) t btain the final frm f the quasigestrphic vrticity and thermdynamic energy equatins. (Equatins and n page 328 f Bluestein). 2 χ = f V g (ζ g f ) f 2 ω (10) p R χ = V g [ ] ω p R (11) Multiply (11) by -(f 2 /) R/p and differentiate with respect t pressure, 4

5 f 2 2 χ = 2 ( ) f 2 ω V g (12a) Nw add (12a) t (10) and rearrange terms t btain the quasigestrphic height tendency equatin. Nte that terms assciated with diabatic effects and frictin have nt been included in (12a) χ = f V g (ζ g f ) R V p ( g ) (12b) Quasigestrphic Height Tendency Equatin Please nte that in many derivatins, equatin (7) is substituted int the temperature advectin term. In that case, the temperature advectin is appxximated by the thickness advectin by the gestrphic wind. Multiply Equatin (11) by (R/p) and then take the Laplacian f the result. Multiply Equatin (10) by by -(f /) and differentiate the result with respect t pressure (/). Subtracting (10) frm (11) and cllecting terms yields the quasigestrphic mega equatin ω = [ ] R 2 V p ( g ) (13) V g (ζ g f ) Quasigestrphic Omega Equatin Equatins (12b) and (13) can be rederived neglecting the assumptin that diabatic effects are minimal. As Bluestein pints ut n pp this assumptin is a bad ne near the surface where diurnal sensible heating/cling effects can be large, at clud tp level when radiatinal effects might be large, and, when latent heat releases assciated with clud develpment χ = f V g (ζ g f ) R p V ( g ) R 1 dq p c p dt Quasigestrphic Height Tendency Equatin With Diabatic Term (14) 5

6 2 2 2 ω = [ ] R p 2 ( V g ) R p 2 1 dq c p dt V g (ζ g f ) Quasigestrphic Omega Equatin With Diabatic Term (15) Interpretatin f the Quasigestrphic Equatins The left hand side f the tw equatins cntains a term that has the appearance f a threedimensinal Laplacian. The term can be estimated analytically, fr example, when the user f the equatin is attempting t evaluate the equatins, say, fr a 500 mb pattern with trughs and ridges. The pattern f mega is als sinusidal. The exact evaluatin will be left t Metr 520 (Dymamic Meterlgy II), and reduces t a cnstant k 2 l f π C (16) p where k and l are cnstants dependent upn wavelength and amplitude f the disturbance at a given pressure level p. Applying (15) t the left hand sides f equatins (13) and (14) allws ne t make the assumptin that χ f V g (ζ g f ) R p V ( g ) R 1 dq p c p dt (17) Simplified Quasigestrphic Height Tendency Equatin With Diabatic Term [ ] R p 2 ( V g ) R p 2 1 dq c p dt ω V g (ζ g f ) Simplified Quasigestrphic Omega Equatin With Diabatic Term (18) Dynamic Effects Thermal Effects The tendency equatin (prgnstic) states that height falls at a given lcatin n a pressure surface (say, 500 mb) when, at that lcatin, psitive (r cyclnic) vrticity advectin ccurs, and/r cld advectin decreasing with height (knwn as differential temperature advectin), and/r sensible chilling decreasing with height (differential diabatic temperature change). 6

7 The mega equatin (diagnstic) states that upward mtin at a given lcatin n a pressure surface (say, 500 mb) is assciated with vrticity advectin becming mre cyclnic with height (knwn as differential vrticity advectin), and/r a warm advectin maximum is lcated at that pint, and/r a sensible heating maximum is lcated at that pint. 7