U.S. DEPARTMENT OF COMMERCENATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION

NATIONAL WEATHER SERVICENATIONAL METEOROLOGICAL CENTER

OFFICE NOTE 192

IsobaricAn Iterative Variational Method for AdjustingWind and Geopotential to Satisfy the Balance Equation

J. P. GerrityR. T. Chu

Development Division

OCTOBER 1978

This is an unreviewed manuscript, primarilyintended for informal exchange of informationamong NMC staff members.

1I- Introduction

If the initial wind and geopotential fields are sufficiently out-of-balance, a primitive equation, prediction model will produce a gravitational-inertial oscillation as a result of the implied need for a "geostrophicadjustment." To reduce the amplitude of such oscillations, one may seek toimpose the "balance-equation" upon the initial data.

The balance-equation may be used to calculate the wind from the geo-potential, or vice versa. In either direction, one may find that thechanges in the initial data are somewhat excessive.

In this note, we discuss a variational method for adjusting both thewind and the geopotential so that the final fields satisfy the balance-equation. The mutual adjustment is carried out so as to minimize the arealaverage of the change in actual and geostrophic kinetic energy. By allowingboth the wind and the geopotential to be modified, one hopes to minimize thedeparturesof both fields from the initial analysis.

2. The Formulation of the Method

In isobaric coordinates, the balance equation may be written

V24 - foV2 - V. - fo)V* + ½V2 (V9'V) = 0, (1)

in which fo is a constant (10 - 4 sec-1), n is the absolute vorticity, 4 isthe absolute vorticity, + is the geopotential, and i is the streamfunction.

Denote by A the ageostrophic terms,

A E - V'(n - fo)Vp + ½V2 ($'V$).i (2)

Now let the analysis yield initial values p* and %* for the stream-function and geopotential. When these are introduced into (1) (with (2)),we may find that,

V2 * - foV2 ~* + A* N* ~ 0 (3)

This means that the initial data are "out-of-balance."

Suppose that we wish to adjust the analysis values so as to obtain abalanced set of streamfunction and geopotential. Let caafid 4c denotechanges in the analyzed values, and define ~' and p' by

= + Oc (4a)

*' := * + Sc (4b)

2

We consider the use of an iterative procedure for effecting a finalbalance. As a first step we try to find lc,' Ac, such that

V2 ' - foV2 ~ ' + A* = 0, (5)

and such that

I MN f[(Vc) 2j (foVPc) 2 ]dxdy (6)

D

is a minimum (where D is the entire domain).

From (3) and (5), we get

V2 c - fcV2 $c + N* = 0 (7)

Since the minimization of (6) is subject to the constraint that (7) besatisfied, we use the Lagrance multiplier X (a function of position, butindependent of Pc and Pc) to involve the constraint (7) into the integral(6):

I = ff[(V4c)2 + (foV4c)2 + X[V2 c - foV2lc + N*j]dxdy. (8)

We consider the variation of I, &I, caused by: a variation in c', 6,Ca variation of Ac, S*c; and a variation of X, 6X. It is necessary that&I vanish if I is to be a minimum.

; +. T x 0. (9)+: ( DP ('x)Since each of the variations 6c, 6kc and 6X is arbitrary and indepen-

dent, it is necessary that

- c 0 (10a)

-oa(&Pc) = : (lb).~~~~~~ (l0b)

= 0 (0). (l

To evaluate the partial derivatives, one proceeds as follows:

Di _lim (I(c + ic Xc ) (c c

(c) 6 c C° consts.c (11)0 6fct~~~~~c' consts. (1

3

I(4c+64C,PC,X) = ff(V~c+VSc) 2 + (foV*c)2 + X[V2~c+V2 Vc-foV2ic+N*]dxdy. (12)

So

I(c+c,*cX) - I(4cfPc,) = ff[2Vfc.*V6c + (V6 c)2 + XV26c]dxdy. (13)

Now if we consider the limiting ratio (11), the term

ff(V6 C ) 2dxdy

may be neglected as being of higher order in the small quantity 6Pc- Thusone is led to require that

ff[2vc -Wv6c + XV2 6c]dxdy = 0 (14)

Similarly one has,

ff[2fo2 V*C.V6pc - AV2 pc]dxdy = 0 (15)

and the constraint equation,

cV2c foV2 c + N* =0. C16)

By invoking the identities,

VJc. V6pc = V-(CS4c Vgc) - c V2 pc

XV2 8¢c = V.(XV6¢c - SpcVX) + 6PcV 2 X

V'C-V 6sc = V-(6qc V4c) - c V2 cc

.V26qc = V .(.Vc - 6cVX) + 6qCV2X

together with Gauss' theorem, one may rewrite (14) and (15) as follows:

,{:#{c[2-Vc- .-VX] + Xi-V6c}di + ff 6c[- 2V24c + V2X]dxdy = 0 (14a)

, i{6c[2foi-Vtc+ i.VX] - foXn.V1c}dJd + ff - 6c[2fo 2 V2 X + foV2 X]dxdy = 0(15a)

The line integrals in 14a and 15a are carried out over the boundary curveenclosing the two-dimensional surface. The unit vector t is the outwardnormal on the boundary curve.

4

For (14a) and (14b) to be satisfied for arbitrarylvalues of Spc and68c, it is sufficient that

1=0

2 n-V4 c = n-V (17)

fo2n-Vpc = - VA

on the boundary curve, and thatin the interior

- 2V24c + v2 X = 0 (18)

2fo . 2 c + V X = 0 (19)

Adding (18) and (19) and using (16), one has

V2X + N* = 0. (20)

One may solve (20) using the homogeneous Dirichlet boundary condition on X(see (17)). With X known, one may satisfy (18), (19) and the remaining

boundary conditions (17), by writing

4 = c X (21)

11-c = x- * (22):2f

Having obtained t*c and cc, one may construct i' and 4<. The lattervalues may then be employed as *, 4* in another iteration of the process.The iteration may be stopped when further steps produce negligibleadditional changes, Ac, or when N* becomes sufficiently small.

The convergence of the iterative process has not been proven. In experi-ments, we have found the method to converge, in the sense that the requiredchanges in 4 become less than 9.8 m2 sec T , after about 10 iterations.

3. The Practical Coding

In the derivation, we have used the streamfunction 4 to represent thewind field. In practice we have employed the wind components themselves toevaluate N*. We have not removed the divergence from the analyzed wind fields,although this could be done if it were found necessary.

5

The computational code uses the following finite-difference formulasto evaluate N* at all interior grid points:

N* m2 (goZ + .5(u2+ v2)) m2 (gZ + 5m2(go (u2+ v2))x

fo~ Y - m2(f y+ ~ _ fo)( X

yxy

- f~Y + m2(fXY + --fo)(U) Y+ 2 =xYf + (f + - 0) Q0 y

with-xyr Y ]x

=m 2 U

and m the map factor of the polar stereographic projection true at 60 0N.

The equation (18) is solved numerically using the SOR method with the "+"stencil for the Laplacian operator. The solution is iterated until X isknown to within 9.8 m2 sec-2.

Having obtained X we calculated the revised values of geopotentialheight Z and wind components using the formulas:

Z = Z + .5 1/go

u u + .5 fo - y

%y-X

v = v - .5 fo x

Note that u, v, and z are unchanged on the boundary gridpoints.

4. Results

Figuresl shows the 300-mb height and wind speed analysis for a portionof the 65 x 65 grid. The height contours are drawn at 60 m intervals; theisotachs are at 10 m s- 1 intervals.

Figure 2 shows the initialized fields after nine iterations. In table 1,we show the maximum absolute value of the height adjustment and the averageabsolute height change for each of the nine iterations.

6

Table 1

Max. Abs. Change (m)

62.8

18.6

9.60

5.47

3.34

2.16

1.46

1.03

.75

Avg. Abs. Change (m)

10.5

3.08

1.21

.53

.25

.13

.07

.04

.02

Figure 3 shows the change in wind speed between the initialized andanalyzed isotachs. The contours are at intervals of 2 m s- 1. The maximumchanges are about 10 m s - 1 (20 knots).

Figure 4 shows the change in the height field between the initializedand analyzed fields. The maximum change is about 70 m. The contourinterval is 20 m.

Figure 5 shows the changes in the height field that would have beenrequired, if one adjusted the analyzed height field to fit the analyzedwind field:through the balance equation. The boundary values of the heightfield were held constant. The contours are at 60 m intervals, and themaximum change is - 125 m.

It seems to us that the initialized fields differ from the analyzedfields by amounts that are within the limits of accuracy of analyses. Ifthis result is replicated in subsequent experiments the use of the methodmay be justified in operational practice.

As developed here, the method is applicable to the 65 x 65 hemisphericgrid. Extension to other grids or to a spectral model appears to betractable.

Iteration

1

2

3

4

5

6

7

8

9

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