RHTEST V4

Buizza and Palmer (1995) suggested that the singular vector growth can be interpreted applying the Eliassen-Palm theorem and the W KBJ method (the letters WKBJ stand for G. Wentzel, H. A. Kramers, L. Brillouin, and H. Jeffreys, who more or less independently discovered the procedure in connection with the solution of different problems), in terms of wave-activity. In this Chapter, first, following Andrews et at. (1987), the Eliassen-Palm theorem is deduced in the quasi-geostrophic approximation on the beta-plane (Section 6.1). Then, the evolution of a wave disturbance in a zonally averaged basic state is analyzed applying WKBJ methods, and wave-activity concepts are introduced (Section 6.2). Finally, these concepts are used to interpret the evolution o f singular vectors computed using the tangent forward and adjoint versions o f the ECMWF primitive equation model.

6.1 The Eliassen-Palm theorem

in the quasi-geostrophic approximation on the beta plane

Consider the equations o f motion on the sphere for the atmospheric flow with a log-pressure coordinate system

z= -H \n .(p lp ) , (6.1)

where p is the surface pressure, J / is a reference height (usually H=7000 m),

and p^ is a standard reference surface pressure (usually p = 1 0 ^ Pa). These equations, in the vicinity of a point with coordinates (Àq, and in the beta- plane formulation, take the form

du

— - / v + ^ =X , (6.2a)

dv _ — + 0 = y , (6.2b) i ? 0 -/fe/c H ^ =— e , (6.2c) * M ’ (6.2d) - r = Q y (6.2e) a t

{Andrews et a i, 1987). 0 denotes the geopotential 0 = f ^ g d Z where Z is

J o Rt c

the geometrical height; d the potential temperature Q = T{p Ip) ^ where R is

the constant for dry air and is the specific heat at constant pressure; x is the eastward distance and y the northward distance defined so that

(ôx,ôy)=([acos4>]ôÀ,aô(l)) with À being the longitude and <p the latitude;

/ = /q+ P 3' (6.3a)

is the Coriolis parameter with/o=2i[2 sincpo and P = 2 ü a 'cos(p(;,

^ ^ d x d y dz^ (m,v,w) = (— , — , — ) ; (6.3b) d t d t d t d d d d d — = — +M— + V + w — ; (6.3c) d t dt dx dy dz

a is the earth radius; X, Y denote unspecified horizontal components of friction; Q is the diabating heating term; p d z ) - p f ^ ^ is a basic density with

p= pJR T ;, partial derivatives with respect to the coordinates are denoted by suffixes.

The Eqs. (6.2a-e) express momentum balance in the zonal and meridional directions, hydrostatic balance in the vertical, continuity o f mass, and the thermodynamic relation between diabatic heating and rate of change of potential temperature.

Denote by Ug=(Ug, 0) the geostrophic wind,

(m ,v ) = ( - i j ; ) , (6.4a)

8 g y X

where

t - / „ ' ( * (6. 4b)

is called the geostrophic stream function and Oj[z) is a suitable reference geopotential profile. Denote the ageostrophic velocity by v^, w j, and decompose the velocity as the sum o f the geostrophic and ageostrophic components,

u = u - u , V = v - v , w =w . (6.5)

a g o g o

Suppose that is a typical order of magnitude o f the geostrophic wind, and that L is a typical horizontal scale. The geostrophic wind is a solution o f the equations of motions under the hypotheses of small Rossby number

Ro=UlfJ^cl, dldt<.fo, pL<fo, and \X\,\Y\<fJJ. The geostrophic wind represents a first order solution o f the primitive equations (6.2).

The quasi-geostrophic equations are the next approximation o f the equations of motion. Substituting Eq. (6.5) into Eqs. (6.2), gives

D u - f V “ P y v =X , (6.6a)

D V + f u - P y u =Y , (6.6b) g g 0 a g ’ (6 6 c) D Q + w Q z = Q , (6.6d) g e a 0 ^ where 0 0 6 , Bzjc H e ^ ^ d - Q j i z ) = H R y ^ e ^ I|f . (6.6f)

The quasi-geostrophic equations can be combined to deduce a single very useful equation, the quasi-geostrophic potential vorticity equation. Denote by Cg the quasi-geostrophic approximation to the beta-plane form o f the vertical component o f the absolute vorticity

■ (6-7)

and by qg the quasi-geostrophic potential vorticity

where

2 7 1 - 1 Rzjc H

e { z ) s Ç N \ z ) , n \ z) ^ H 'RQ^(z)e ' . (6.9)

Eliminating the vertical ageostrophic wind component from the vorticity equation that can be deduced from the momentum equations, gives the quasi-

geostrophic potential vorticity equation.

Consider now the problem of describing the evolution of a small disturbance of a zonal mean basic state. Denote with an overline zonally

averaged quantities, e.g. u = — . Denote by « =(w,0,0)

2 n J o

a geostrophic zonal basic state which satisfies the geostrophic equations

w = -ip^ , (6.11a)

0 - 0 { . z ) = H R ' ^ f e ' ip . (6.11b)

0 0 V

Rzjc H

By substituting u = u + u ' in the Eq. (6.10), the linearized version of the

quasi-geostrophic potential vorticity equation can be deduced:

D q ' ^ v ' q =Z ' , (6.12a) with ' ( 6 '2 b ) where - d _ 6 D = — +M— (6.13) dt dx

is the disturbance potential vorticity, and

« , " P - \ - P o ' ( P o ^ “z \ (6.15)

is the basic state northward potential vorticity gradient.

Multiplying Eq. (6.12a) by 9qQ I ^ » taking p

Jq

neglecting , and zonally averaging, the following equation can be obtained

a 1

z 'q

g ^ ( 2 p o T ^ " ^ ' ' ^ = p o - r ’

y y

where F is the E liassen-Palm flux. Equation (6.16) is also called the Eliassen-Palm theorem. In the quasi-geostrophic beta-plane approximation the Eliassen-Palm flux F is given by

F = ( 0 ,- p ^ v M ^,p^^vV /0^) , (6.17a)

or, since u v^ = i|f

and the divergence of the Eliassen-Palm flux can be written as

1

The term A = — p , proportional to the wave potential enstrophy

2 ^ q y

I - J2

—q ,is called wave-activity density. In case of no diabatic heating and

zero non-conservative forcing, X=Y=Q=0, the time derivative of the wave- activity density depends on the divergence of the Eliassen-Palm flux.

6.2 Wave propagation under WKBJ theory

WKBJ or Liouville-Green methods suppose that each disturbance quantity, such as the disturbance zonal velocity u', can be written in the form

« ' = R e [ « ( ; r , O e , (6.18)

where x=(x,y,z) and the phase % is real. Define the local wave number vector

k=(k,l,m) and the frequency o) as

/ = = w = • (6.19)

dx dy dz dt

2 n 2ti

Denote by 1 = , T = — the typical length and time scales of the

phase %, and by L, T the typical length and time scales o f u, k, /, m, (o .

WKBJ or Liouville-Green methods suppose that m, k, /, m, O) and the basic

state characteristics vary much more slowly in time and space than does the phase %, i.e. that

- L - T

L = 0 (— ), T=0 {— ) w h e r e i s a s m a l l WKBJ p a r a m e t e r . In o t h e r

w o r d s , M, ky /, m, co v a r y o n l e n g t h a n d t i m e s c a l e s m u c h l o n g e r t h a n t h e

p h a s e .

By substituting Eq. (6.18) into the linearized equation of motion for the disturbance, a dispersion relation can be obtained.

( ù = A ( k ; x , t ) . (6.20)

The wave group velocity is defined as