(! g. + f ) Dt. $% (" g. + f ) + f

Transcription

1 The QG Vorticity equation The complete derivation of the QG vorticity equation can be found in Chapter Please read this section and keep in mind that extra approximations have been made besides those that lead to the vorticity equation 4.22). Those approximations include i. V a! V or V a / V " ORo) ii. iii. The advection velocity is eostrophic f! + "y i.e. midlatitude beta-plane approximation The resultant QG vorticity equation is or D! + f ) Dt = " # #p!"!t! = #V " + f ) + 6.9) In Cartesian coordinates the eostrophic wind with constant-f) is defined as V! " k # Thus the eostrophic vorticity! = k "# V can be expressed as! = "v "x # "u "y = " 2 "x + "2 ' 2 "y 2 ) = ) For a pure 2-D motion the vorticity equation can be written as!!t "2 # =!#!!y!x +!#! '!x!y) "2 # + f ) 4.28 ) For a 3-D QG flow we have to deal with the stretchin term as well we will demonstrate how to express the conservation of QG PV in terms of eopotential heiht in the QG PV equation from pae 5).

2 Example of vorticity advection for a 2-D case 4.28 ) can be rewritten as!v "# + f ) =!V " #! v The two term on the rhs represent the eostrophic advections of relative vorticity and planetary vorticity respectively. For disturbances in the westerlies these two effects tend to have opposite sins. Advection of relative vorticity tends to move the vorticity pattern and hence the trouhs and rides eastward downstream). However the advection of planetary vorticity tends to move the trouhs and rides westward aainst the advectin wind field. The latter motion is called retrorade motion or retroression. The net effect of advection on the evolution of the wave pattern depends on the scale of the wave perturbations. To demonstrate the point we consider an idealized eopotential distribution on a beta-plane consistin of the sum of a zonally averaed part which depends linearly on y and a zonally varyin part wave part representin a synoptic wave disturbance) that has a sinusoidal dependence in x and y:!x y) =! 0 " Uy + Asin kx cosly 6.20) 2

3 Here y = a! "! 0 ). Then the correspondin eosstrophic wind are iven by u =! "# "y v = "# "x = U + lasin kx sinly = kacos kx cosly The eostrophic vorticity is then simply! = " # 2 = "k 2 + l 2 )Asin kx cosly With the aid of these relations it can be shown that in this simple case the advection of relative vorticity by the wave component of the eostrophic wind vanishes and so that the advection of the relative vorticity is "#!u "x! v "# "y =!U "# "x = +kuk 2 + l 2 )Acos kx cosly While the advection of the planetary vorticity is Therefore if k 2 + l 2!"v =!"kacos kx cosly ) <! /U the synoptic wave should move westward retroression) ) >! /U the wave should move eastward. And waves of intermediate ) =! /U can be stationary. and if k 2 + l 2 wavelenth i.e. k 2 + l 2 3

4 The 3-D dynamics overned by the conservation of QG PV is far more complex a topic to be discussed next. 4

5 The QG Potential Vorticity equation 3D) The QG PV equation is derived from utilizin both the QG vorticity equation and the thermodynamic equation. From the homework of Chapter 2 " #!T!t + u!t!x + v!t!y Recall that! " #w and that d # ) / = S p # T ' ' p So ' + w d ) ) = J 2.55 ) C p!t!t + V " # T ' h ) + p ' R ) = J 6.3a) C p where! " #RT 0 p # d ln 0 / dp and! 0 is the potential temperature correspondin to the basic state temperature T 0.! " 2.5 # 0 6 m 2 Pa 2 s 2 in the mid-troposphere. From 6.2) that is 6.3a) can be expressed as!" = # = # RT p!!t + V ' " # h )!+ ' ) - =. J p 6.3b) Multiplyin 6.3b) throuh by! and differentiatin with respect to p yields:!!# ' " ) +! - " V!' 0!3 / + ) 2 +. = 4 f! 5 J ' 0 " p ) 6.22) where we use! to denote!"!t and! " R / C p. 5

6 While the QG vorticity equation can be written in terms of eopotential tendency as follows.! 2 " + V #! '! 2 + f ) = + +p!" Substitutin it in 6.22) throuh we have +! 2 + " "p # for the adiabatic motion. " ' - "p ). / 0 + V! f! f ' ) + " "p # V! "2 ' - "p ) / = ) 0 +. Notin that! = ) the equation above can be reoranized as!!t + " 2 +! #! ' - ) /0 + V " " f +!. + #!0 ' - ) / +. #!V!0' " ) = 0 However for the thermal wind relation!v / = k " #! / ) so the last term drops off the equation and we have finally:!!t + V ' "# ) q = D q = ) Dt where q is the quasi-eostrophic potential equation defined by q! " 2 # + f + p ' # p ) ) For diabatic flow the QG PV equation should be D q " # J =! Dt "p ' p) 6

7 Geopotential tendency 6.23) is often referred to the eopotential tendency equation as it can be used for the purpose of dianosin the tendency of the eopotential.! 2 + " 2 " ' - "p # "p ) / 0 = V 2! '! f +!###" ###. f 0 ) " "p f 2 0 # V 2! "3 ' - "p ) /!#### "####!#### + "####. 6.23) A B C A: local eopotential tendency B: vorticity advection C: vertical differential thickness advection Note the! 2 " # ". So equation 6.23) provides an immediate qualitative implication for the tendency of eopotential with known distribution of!. Term B is usually the dominant forcin term in the upper troposphere wherein a risin fallin) eopotential is associated with neative positive) advection of absolute vorticity. Note that the advection term as its name implies does not chane the strenth of the disturbance at the levels where the advection is occurrin but only acts to propaate the disturbance horizontally and to spread it vertically as will be shown later in the lecture. Term C represents a major mechanism for amplification or decay of midlatitude synoptic system. Term C is also proportional to the minus the rate of chane of temperature advection with respect to pressure i.e. plus the rate of chane wrt heiht). It is also referred to as differential temperature advection. Differential temperature advection enhances upper level heiht anomalies in developin disturbances. Below the 500-hPa ride there is stron warm advection associated with the warm front whereas below the 500-hPa trouh there is stron cold advection associated with the cold front. The former increases thickness thus builds the upper level ride; the later decreases the thickness thus deepens the upper level trouh. Above the 500-hPa level the temperature radient is usually flatter advection tends to be small. Thus in contrast to Term B in 6.23) the effect of forcin term C is concentrated in the lower troposphere. In the reion of warm advection!v "#! / p) > 0 as V as a component down the temperature radient. But the advection also decrease with heiht therefore 7

8 ! '"V #"! / ) ) / > 0. Conversely beneath the 500-hPa trouh there is cold advection decreasin with heiht that ives rise to deepenin effect on eopotential.! = "# "t 3 5 " -! "p V ' "# 0 4 / ) "p > 0 at the ride < 0 at the trouh 8

9 The temperature advection pattern described above indirectly implies conversion of potential enery to kinetic enery. 9