THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Atmospheric and Oceanic Science Technical Report RESEARCH NOTES Aksel C. Wiin-Nielsen Project Director DRDA Project 002630 supported by: NATIONAL SCIENCE FOUNDATION GRANT NO. GA-16166 WASHINGTON, D.C. administered through: DIVISION OF RESEARCH DEVELOPMENT AND ADMINISTRATION ANN ARBOR May 1974

TABLE OF CONTENTS Page A NOTE ON THE MOTION OF INERTIAL WAVES ON THE SPHERE 1 A NOTE ON BAROCLINIC INSTABILITY AS A FUNCTION OF THE VERTICAL WIND PROFILE 9 A NOTE ON FJORTOFT'S BLOCKING THEOREM 19 A NOTE ON THE ANGULAR MOMENTUM BALANCE OF THE ATMOSPHERE 33 iii

A NOTE ON THE MOTION OF INERTIAL WAVES ON THE SPHERE by A. Wiin-Nielsen Department of Atmospheric and Oceanic Science The University of Michigan Abstract The motion of inertial waves on the spherical earth is found as a function of the zonal wave number by using a perturbation method with a basic state of no motion. The speed of the waves is compared with the elementary wave formula, derived under the assumption of a constant Coriolis parameter. The inertial waves can be investigated as a special case of a more general investigation, conducted earlier by the author, of transient waves in the atmosphere. 1

1. Introduction The motion of inertial waves is generally described by the well-known formula ~ f/jk where fo is a constant value of the Coriolis parameter while k is the wave number. It is obvious that the formula is valid formally for short waves only because no variation of the Coriolis parameter is permitted. Inertial trajectories on the spherical earth have recently been investigated by Wiin-Nielsen (1973) showing marked deviations from the inertial circle for large initial wind speeds and/or initial positions in the very low latitudes. It would appear worthwhile to investigate if corresponding differences are found between the simple wave formula given above and speeds computed without the simplifying assumptions. The author (Wiin-Nielsen, 1971) has investigated the general problem of the motion of the vertical modes of transient waves in an atmospheric basic state characterized by no motion, but an arbitrary thermal stratification. It was found that the speed of the waves is determined by three coupled, ordinary differential equations involving the streamfunction, the velocity potential and the geopotential for a given vertical mode of the perturbation. The inertial waves are characterized as the waves which would exist if there were no pressure field. It follows therefore that we may obtain the equations governing the motion of inertial waves by simply disregarding all reference to the geopotential of the perturbation. In this way we obtain two coupled equations which can be solved by using a numerical technique analogous to the one employed in the earlier investigation. 2. Problem and Procedures The perturbation equations for inertial waves on a basic state of no motion are __ = fv at )v a = -flu (2.1) where f = 2 sin cp. (u,v) is the horizontal velocity vector, u the zonal velocity, v the meridional velocity, t time, Q the angular velocity of the earth, and cp latitude. 2

The system (2.1) is replaced by the vorticity and the divergence equations: - 22 = -fV v - cos cp v a^~~t a O7 - 2Q/ a7t*V = 0- cos cp u (2.2) at = f~- a where 7 = V2, V * v = V2x and = 17 V = [ a + 1 (Os m (2.3) 2V = 2 2s 2r 2 2 cos cp 6 p C 1 a a Lcos cp in which a is the radius of the earth, X the longitude, ~ the vorticity, V * v the divergence,r the streamfunction, and X the velocity potential. Note, that 1 at 1 ax a Acp a cos cp c v. +ilax (2.4) a cos cp b a ap We notice that all the dependent variables can be expressed in terms of the streamfunction and the velocity potential. The perturbations are expressed in the form im( - ct) = m(T) e X = iXm(p) eim( t) (2.5) Introducing the nondimensional phase speed s by the relation v mc s = - 2 - - 2 (2.6) 2where v is 2 n where v is the frequency, we find after substitution of (2.5) in (2.2) that 3

2 m 2 sV 7 rm + mlm +(1 - ) + s x sm m 2 )y+ x 2 s V2 X + mX + (1-1 2) a + V2 = s m L s m in which p. = sin cp. '0 0 (2.7) The problem is now to which s is the eigenValue. eigenvalues by expanding 'm solve the eigenvalue problem presented by (2.7) in We obtain a numerical procedure to find the desired and Xm in series of Legendre functions, i.e., R a (p) = Z a(m, m+2r) P(m, m+2r, p) m r=O or a ( l) R = Z a(m, m+2r+l) P(m, m+2r+l, p) r=0 (2.8) in which R = rmax denotes the truncation in the series (2.8), and where the first and second forms are used for even and odd functions, respectively, (2.8) are substituted in (2.7). Making use of a number of identities for the set of Legendre functions we can reduce the resulting equations to the following pair for each value of r: (2r+l)(m + 2r) A + (m+ 2r+l)(m+ 4r + 1) x(mm+2r) m (m + 2r +l)(m+2 r +2) - s) (m, m+ 2r +l) (m+2r+3)(2m+2r+2) A, (m+ 2r + 2)(2 +4r + ) m m + 2r + 2) (2r(2m + 2r -l) (m + 2r)(2m + 4r - 1) m.m+2r-1 =0 m+m+ + (m + 2r)(2mn + 2r - 1) - s AX(m, m+2r) x~ (m + 2r + 2) (2rn + 2r + 1) Am (m+2r+2)(2m+2r+3) (m m+2r - (m+'2r +)an(+r+ + ) m++l) (2.9) We notice from (2.9) that even functions for the coupled with odd functions for the velocity potential given value of m the parameter r will run through the streamfunction are and vice versa. For a values 0,1,2,...,R. We

have thus 2(R +1) equations which leads to finding tne eigenvalues in a standard eigenvalue matrix. It is not obvious at the outset how large R must be in order to determine the largest eigenvalues tith sufficient accuracy. This problem was solved experimentally by gradually increasing R in steps of 1 until no change was found in the largest eigenvalues. Experience showed that R in no case needed to exceed 20. 3. Results The system (2.9) was solved as described above for a fixed value of m, selecting R in such a way that we have a good accuracy for the largest eigenvalues. Using this procedure we can repeat the calculation for a set of values of m and thereby get the phase speed as a function of the zonal wave number. These calculations were carried through for the values m = 1,2,...,15. For each value of m we have determined the first six eigenvalues. As one would expect from the simple theory with a constant Coriolis parameter the eigenvalues appear in pairs witn approximately the same absolute value, but of opposite sign. Table I gives the numerical values of c, expressed in the unit: deg day1, for the first six eigenvalues. The computed values of s were converted to c using (2.6) and a value of Q = 360 deg day-1. It is seen that c decreases in absolute value as m increases for each of the six eigenvalues. TABLE I NUMERICAL VALUES OF THE FIRST SIX EIGENVALUES (Expressed in the unit: deg day-1) m c1 c2 c3 C4 c5 c6 1 -720.14 716.76 -714.15 708.27 -703.97 2 -359.58 357.20 -356.43 352.25 -351.51 345.57 3 -239.05 237.27 -236.44 233.54 -232.69 228.73 4 -178.72 177.27 -176.42 174.17 -173.30 170.31 5 -142.49 141.23 -14o.38 138.52 -137.68 135.25 6 -118.30 117.19 -116.35 114.75 -113.92 111.89 7 -101. 00 100.00 - 99.16 97.77 - 96.96 95.20 8 - 88.01 87.10 - 86.27 85.03 - 84.23 82.69 9 - 77.89 77.06 - 76.24 75.12 - 74.34 72.97 10 - 69.79 69.02 - 68.22 67.20 - 66.43 65.19 11 - 63.16 62.44 - 61.65 60.71 - 59.97 58.84 12 - 57.63 56.96 - 56.18 55531 - 54.58 53.55 13 - 52.94 52.31 - 51.54 50.74 - 50.03 49.08 14 - 48.93 48.33 - 47.58 46.83 - 46.13 45.25 15 - 45.45 44.88 - 44.14 43.44 - 42.76 41.94 5

One of the first problems to investigate is if the values in Table I agree reasonably well with the simple formula + fo/k.. In order to make such a comparison we write f 0 C = k 2 sin cp0 ~ L 2it 2 f sin cp 2xc 360~ m 720~ sin p m (5.1) where we have expressed the wavelength in degrees of longitude and O = 2t day1. It is obvious that if (3.1) shall give values in agreement with the numerical values in Table I we must select cp = 1/2A. The solid curve in Figure 1 displays the relation c = 720/m, while the circles give the values of c6 from Table I. It is obvious that the values I cllc21,..., c51 would be even closer. We have therefore shown empirically that the phase speed of inertial waves are well approximated by a formula where the phase speed is inversely proportional to the zonal wave number. This relation is of course not exact. If it were, we would, according to (2.6), have s = 1 as the eigenvalue. The numerical determination of s shows, however, that the largest eigenvalue s1 deviate only slightly from unity as shown in Table II. TABLE II THE LARGEST EIGENVALUE sl AS A FUNCTION OF m m 1 2 3 4 5 6 7 8 S1 0.9978 0.9964 0.9936 0.9905 0.9871 0.9834 0.9795 0.9755 9 10 11 12 13 14 15 0.9713 0.9670 0.9626 0.9581 0.9536 0.9490 0.9445 4. Concluding Remarks The speed of inertial waves on the sphere has been calculated. The results show that the various eigenvalues, ordered according to their absolute value for the same value of the longitudinal wave number, decrease relatively little. It is also shown that the phase speed of the inertial waves is, for practical purposes inversely proportional to the longitudinal wave number. 6

800 I I I I I I I I I I I I I I I 700 600 - 500 1 I 0 0) 400 (D U 300 H 200 H 100 h I I -- - I I I I I I I I I I I I 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 m Figure 1. The curve shows the relation c = 720/m while the circles indicate the values C6 taken from Table I. 7

It is naturally possible to calculate the functions rm(p) and Xm(p) by summing the series (2.8) after having found the eigenvector R(m, m+2r) and x(m, m+2r) from the solution of the system (2.9). Such calculations have been made, but the results are not reproduced here because no significant conclusions have been drawn from the results. 5. Acknowledgment The research leading to this paper has been supported by the National Science Foundation under Grant No. GA-16166. Mr. James Pfaendtner has been responsible for the programming of the numerical calculations. References Wiin-Nielsen, A., 1971: On the motion of various vertical modes of transient, very long waves, Part II, the spherical case, Tellus, Vol. 23, No. 3, pp. 207-217. Wiin-Nielsen, A., 1973: On the inertial flow on the sphere, Technical Report, The University of Michigan, 002630-6-T, 52 pp. 8

A NOTE ON BAROCLINIC INSTABILITY AS A FUNCTION OF THE VERTICAL WIND PROFILE by A. Wiin-Nielsen Department of Atomspheric and Oceanic Science The University of Michigan Abs tract The note provides an additional example of a solution of the quasigeostrophic, baroclinic stability problem in an atmosphere with adiabatic stratification. The main purpose of the example is to consider wind profiles in which the maximum may vary in position. For each profile it is possible to determine the region of instability in a diagram with the maximum wind as ordinate and the wavelength as abscissa. In additon, the degree of instability, measured by either the imaginary part of the wave speed or by the e-folding time, can be calculated. 9

1. Introduction A few years ago the author gave a solution of the quasi-geostrophic, baroclinic stability problem in an atmosphere with adiabatic, vertical stratification with a arbitrary vertical profile of the horizontal wind in the basic state (Wiin-Nielsen, 1967). Several examples were given in the original paper. The purpose of this note is to provide an additional example using a wind profile which is rather general. 2. Review of the Solution The eigenvalue problem for the adiabatic, quasi-geostrophic stability question may be stated by the following equation for the (complex) amplitude of the perturbation "vertical velocity" t = Q(p ), where p* is a nondimensional pressure, p,.= P/pO, p is pressure and p = 100 cb: 2 d 2 dE d2 E(E - c)~ - (2E - c) -a 0 dp2 dp dp - (2.1) where E = U(p*) - c, U = U(p*) speed, cr = P/k2, p the Rossby turbations are of the form the given wind profile, c the (complex) phase parameter, k the wave number and where the per cI = W(p ) exp[ik(x - ct)] (2. The solution to (2.1) was given in the previous paper by the expression 1 c = (I+ 2 I + 12) (2.7 1 2R - 1 2 4R?) 5) in which (2.4) I = i U dp and I2 =o1 (2.5) 10

3. A New Example We shall prescribe the vertical profile of the horizontal wind by the expression U (p (n - - 1 ): r - n- r - r 1 UN( = r -1)!(n - r - P* (1 - p) (3.1) where UN = U/Um, Um is the vertical average of the wind, i.e., Um = I, P* is the nondimensional pressure, while r and n are parameters which determine the shape of the wind profile. It is seen that (3.1) is closely related to the well-known beta function, and that the expression in the first fraction guarantees that o UN (P) dp. = 1 (3.2) Some selected examples of the vertical wind profiles given by (3.1) are shown in Figure 1. It is seen that the parameter n, for a fixed value of r, determines the position of the maximum in such a way that increasing values of n corresponds to a maximum wind at lower values of the pressure. As seen from Figure 1 it is possible to produce many shapes of the wind profile by selecting r and n in a suitable way. For example, n = 2r will produce a wind profile which has its maximum at p. = 0.5 and is symmetric around this level. (r,n) = (2,4) (3,6), (4,8), and (5,10) are examples of such profiles. We note furthermore that if we in (3.1) replace r by n - rl, but keep n the same we get a profile U(p),. =. - (n - 1) )r - N * (n- r - l):(r - 1)' P* ( ) Introducing for convenience the variable p. = s + 1/2 we find (3.1) to be U (s) n = f 1)!.1 + - (3.4) Ns- (r -l).(n - r - 1). 2 2 while (3.2) becomes UN(S) - 1 sn - 1)'nrl-1 - i) rl(- 1 =(n- r1 -1)'(r1 - ^ 1 - ) 1 (3.5) 11

02 04p 06 1 2 3 r=2,n=4 0-8 1.0 3 r=2,n=5 r =3,n=6 r=5,n=8 0-2 0-4 06 0-8 r= 6,n=8 1 2 r=6,n=10 1 2 r=6,n=11 Figure 1. Examples of wind profiles for various values of (r,n). The curves have nondimensional pressure as the ordinate and the nondimensional basic current as abscissa. 12

The variable s goes from the value -1/2 at the top of the atmosphere, 0 at p = 1/2, to the value +1/2 at p = 1. Replacing s by -s in (3.5) it is seen that (3.5) goes into (3.4). The profiles' (3.1) and (3.2) are therefore symmetric around p* = 1/2, and they have the same values of Il and 12 which implies that the complex phase speed is the same for the two profiles. In particular, we have shown that the rate of instability is the same for wind profiles which are symmetric around the middle of the atmosphere. The result would apparently not be true if the stratification was different from the adiabatic structure in the present model. The profiles shown in Figure 1 show several examples of such symmetric profiles such as (r,n), = (2,5) and (r,n) = (3,5), (r,n) = (2,6) and (r,n) = (4,6), (r,n) = (2,7) and (r,n) = (5,7) and several others. The nondimensional phase speed, c = c/U for the profile (3.1) can be evaluated from (2.3). We find that Il7Um = 1 and, after a simple integration, ( "\2~2 2 n - ) (2r - 2):(2n - 2r - 2).' F(r,n) = = (r - l)=(n-r - l)!( (2n -) (3.6) The result is C. = (1 L-i) - l F(rn) + 64- (3.7) \,1 Um 8t2/ U- 6 4Tr When the quantity under the square root is negative, we have instability. In this case we find 2 c = 1- L (3.8) *r U 8T2 and L 4 1/2 c = Ffrn) -1- - - 6 (3.9) which shows that the speed of the unstable waves are independent of the parameters r and n, but dependent on the vertically averaged speed Um. (3.8) gives the well-known result that the speed of unstable waves in the present model is very similar to the speed of Rossby waves in a barotropic, nondivergent atmosphere. (3.9) shows that sufficiently short waves are always unstable. The critical wavelength below which the waves are unstable is 13

c - J ( Y^rn)! (3.10) L = 2/ Tc( m)l/2 (F(rn) - )l/+ (5.l) which may also be written in the form B 1 2 U = 3-) (3.11) m 8 2 ci(O) c where ci(0) = (F(r,n) - 1)1/2 (3.12) is the imaginary part of the phase speed at L = 0. It is seen from (3.11) that the region of instability is inside a parabola in a diagram with L as the abscissa and Up as the ordinate. The region of instability is large if the coefficient to L2 is small. The largest region of instability will be obtained for the maximum value of c i(o). Because of the fact that the instability is the same for two profiles which are symmetric around p= 1/2 we need consider only those profiles for which the maximum occurs at a nondimensional pressure p* < 1/2. It is found by differentiation of (3.1) that dUN (n - 1) r-2 -2 dp- (r - 1)!(n- r- 1) r - 1) (n 2)p] p. ( p-J (3.13) which shows that UN has its maximum at a value of p* equal to r 1 P. m = (314) P,*m n - 2 (3.14) We find pm < 1/2 when r < (1/2)n in agreement with the examples shown in Figure 1. It is of course common practice to illustrate the region of instability in a diagram with the windshear as ordinate and the wavelength as abscissa. We may define a windshear Us by the formula Um 1 dUN) Um U -1- P.- mdp =* U (p (53.15) *.~~~~m P*IM *.Ym~~~~~~~ 14

Using (3.14) and the definition of UN we find s (n - 1)' (r )r- (n- r l)n-r-2 Um (r- 1)(n- r- 1)' (n-2)n(n- 2) and (3.11) becomes 2 U = (r,n) L (3.17) 5 C where r-l n-r-2 (n- 1) ' (r - ) - r (3.18) ( - 2) 8-2 C*i(O) A calculation of y(r,n) for various values of (r,n) shows the general result that the region of instability in a (L,US) plane becomes larger the closer the wind maximum is to any one of the boundaries p* = 0 and p* = 1. For a given value of r we find therefore the smallest region of instability when n = 2r, i.e., for the wind profile which is symmetric around p* = 1/2. On the other hand, among these symmetric profiles we find the largest region of instability when n = 2r is the largest, i.e., when we have the sharpest maximum. This result is in agreement with those obtained in the previous investigation. In order to illustrate the degree of instability we have prepared Figure 2 which shows the e-folding time as a function of wavelength for the selections (r,n) = (2,4), (2,5)...,(2,8). These curves show that the degree of instability increases as the maximum of the wind profile approaches the upper boundary. Similar curves are naturally obtained if we let the wind maximum approach the lower boundary. Figure 3 illustrates the degree of instability, measured by the e-folding time as a function of wavelength, for the wind profiles which are symmetric around p* = 1/2. It is seen that the degree of instability increases as n = 2r increases, i.e., the sharper the wind maximum becomes. 15

T,days (2,4)(2,5) (2,6) (2,7) 3 2 0 1 2 3 4 5 6 7 8 9 10 L,106m Figure 2. The e-folding time in days as a function of the wavelength in 106m for the values (r,n) = (2,4),(2,5),...,(2,8). 16

T,days (2,4) (3,6) (4,8) 3 2 1 1 2 3 4 5 6 7 8 9 10 L,106m Figure 3. As Figure 2, but for the values (r,n) = (2,4), (3,6), (4,8), and (5,10). 17

4. Concluding Remarks The purpose of this note has been to investigate the region and the degree of instability as a function of the vertical profile of the horizontal wind. In the quasi-geostrophic model with adiabatic stratification it is found that (a) the instability is the same for two wind profiles which are symmetric around the middle of the atmosphere. A low level jet will thus show the same instability as a high level jet; (b) the instability becomes larger the closer the jet is to either of the boundaries; and (c) among the symmetric wind profiles those with the sharpest maximum will be the most unstable. Reference Wiin-Nielsen, A., 1967: On baroclinic instability as a function of the vertical profile of the zonal wind, Monthly Weather Review, Vol. 95, pp. 733-739.

A NOTE ON FJORTOFT'S BLOCKING THEOREM by A. Wiin-Nielsen Department of Atmospheric and Oceanic Science The University of Michigan Abstract The changes of kinetic energy and enstrophy in a three-component system in a barotropic, nondivergent fluid are analysed in detail. For a given change of kinetic energy or enstrophy on the intermediate scale, the changes on the large and the small scale components are calculated. The conditions under which the change on the large-scale component is larger than the change on the small-scale component are found for both enstrophy and kinetic energy as a function of the latitudinal scale parameter. 19

1. Introduction Fj^rtoft's (1953) theorem states that kinetic energy must be transferred toward both larger and smaller scales in a two-dimensional, nondivergent flow. This property is due to the conservation of both kinetic energy and enstrophy in a barotropic flow. The theorem is most easily proven for a low-order system consisting of three components only. For such a system one can easily calculate the changes in the kinetic energy of the largest and the smallest scales for a given change of kinetic energy on the middle scale as it was done by Fjfrtoft (loc. cit.). In atmospheric flow we have on the average a maximum conversion of available potential energy to kinetic energy on a scale around longitudinal wave number 5-8. If Fj^rtoft's theorem is applicable to the baroclinic atmospheric flow, one should expect a nonlinear transfer of kinetic energy from this scale to both larger and smaller scales. Data studies, summarized by Steinberg et al. (1971), show that such transfers can be computed from atmospheric data with a result in agreement with the theorem. It has on occasion been stated that Fjartoft's theorem implies a larger transfer of kinetic energy to the large scale than to the small scale. This implication is perhaps due to a printing mistake in the original paper on the subject or to an erroneous use of a mechanical analogy suggested by Charney (1966). It appears that observational studies (Steinberg et al. (1971), Saltzman (1970), Saltzman and Teweles (1964)) verify the statement, but it must be kept in mind that these observational studies are arranged according to the longitudinal wave numbers, while the theory is developed for spherical harmonics, and that rather arbitrary divisions in three wave groups have been used in these studies. In view of the facts mentioned above it seems desirable to investigate the conditions under which a larger transfer of kinetic energy goes to the larger scales than to the smaller scales. The question can be answered by elementary calculations as long as we restrict ourselves to a three-component system. Such an analysis is given in the following section combined with a similar analysis for enstrophy changes. 2. Three-Component System Let us assume that we consider an interactive three-component system. By this we mean that the longitudinal wave numbers m satisfy the selection rule as given by Platzman (1960), and that the meridional scale parameters nl < n2 < n3 satisfy the selection rule 20

n3 - n < n2 < n +n3 (2.1) or, equivalently, n2 -n < n 2 < n3 < nl + n3 (2.2) Denoting q = n(n +1) we may strophy by the relations AK + AK + AK3 qlAK + q2AK2 + q3 express the conservation of energy and en= 0 =0 (2.3) energy on the intermediate scale we may Assuming a change AK2 of kinetic find AK1 and AK3 from (2.3) giving AK1 AK3 q3 - 2 = - q'lAK2 q2 -q q - - ---- lAK2 cj -\ 2 (2.4) Since each of the fractions in find that AK1 and AK3 have the same easy to show that 3 -q2 0 < '' < 1 3 - ql (2.4) is positive because ql < q2 < q3 we sign, opposite to the sign of AK2. It is (2.5) < JAK21. We shall next investigate the Lrger than 1. We find from (2.4) that and q2 - ql 0 <- < 1 3 - q2 showing that IAK11 < IAK21 and [AK31 conditions under which AK1/AK3 is la 21

AK1 AK3...... q3 - 2 q2 - cl and it follows therefore that AK if a> 1 if and only if (2.6) (2.7) q2 <(1/2)(q +q 3) -= A 9 (2.8) where qA is the arithmetric mean value of ql and q3. We may also express (2.3) in terms of the enstrophy for the components. We have then AE + AE 1 5 - AE2 1 1 - AE + - AE ql 1 i3 3 which may be solved giving q2 (2.9) AE1 AE3 = — 1..3.. AE i. 2 2 -c 1 3 2 1 c2 c3 -ql 2 (2.10) (2.10) may naturally be obtained directly from (2.4) by using the relationship AE = qa-2 AK where a is the radius of the earth. (2.10) shows that AE1 and AE3 have the same sign, opposite to the sign of AE2. We should now like to consider the conditions under which AE1/AE3 is larger than unity. We find that this is the case if and only if q q < ( 2)(1 ) 21 J2 (ci +ci ) 153 2 qG A % (2.11) 22

where qG, the geometrical mean, is defined by the relation qG = (qlq3) /2 It is well-known that qA > qG. From this relation it follows that qA > q/qA' In view of this relation it is seen that if AE1/AE3 > 1, then it is also true that AK1/AK3 > 1. Conversely, we may say that if AK1/AK3 < 1 it follows that AE1/AE3 < 1. The condition which corresponds to the results of observational studies, i.e., AK/AK3 > 1 and AE1/AE3 < 1, will take place in a three-component system only when q/qA < q2 < qA' These selection rules can easily be used to test any three-component system which has been selected for integration. If the system shall be active by which we mean that the interaction coefficient has a non-zero value we must also according to Platzman (1960) have n3 - n < n2 < n3 (2.12) Suppose that we want to find the active systems for which AE/AEE3 > 1. Denoting'q* = q./qA we require 2 < qn2 + n - < 0 (2.15) @2 < @*' 2 2 q* (2.13) 2 < q* n2 2 which is satisfied if n2 < n = (1/4 + q )/2 1/2 (2.14) In addition, we must satisfy (2.12). It is straightforward to show that n* < n3 in all cases. The final condition for AE1/AE3 > 1 is therefore n -n < n 2 < n (2.15) (2.15) gives the lower and upper limit for n2 necessary to insure that more enstrophy goes to the larger scale than to the smaller scale. These limits are easily calculated for a given pair (nl,n3). It is, however also seen from (2.15) that in order to find suitable values of n2, it is a necessary condition that n3 - n < n (2.16) 23

(2.16) shows that the choice of (nl,n3) is restricted to a certain region in the first quadrant of the (nl,n3) plane. We shall next determine this region. Substituting from (2.14) in (2.16) and using the definitions of q., qA and qG we may after considerable algebraic manipulations show that (2.16) is equivalent to the following inequality: F(n n3) < 0 (2.17) where F(nn3n) = nl(n1 +l)(n1 -2n3-1) - n3(n3 +1) (2n -n) (2.18) To determine the region in which (2.17) is satisfied it is most convenient to find the roots of F(nl,n3) = 0 in the region n3 > nl. This was done by writing F(nl,n3) as a 4th degree polynomial in n3, determine the roots-of the polynomial for a fixed value of nl, and repeat the process by selecting a sufficient number of values of nl. The results of these calculations are shown in Figure 1 in which the area between the curves nl = n3 and F(nl,n3) is the region in which F(nl,n3) < 0. For any pair of values of nl and n3 selected within the region we can calculate the lower limit, i.e., n3 - nl, and the upper limit, i.e., n., for n2 necessary to obtain AEjAE3 > 1. It should be kept in mind that n2 has to be an integer. n3 - nl and n* may be so close to each other that no integer can be found between them. Table I shows all the triplets (nl,n2,n3) for which AE1/AE3 > 1. For each entry (nl,n3) for which 1 < n1 < 17 and 1 < n3 < 20 we have listed the permissible values of n2 for which AE1/AE3 > 1. Let us next turn our attention to the kinetic energy and find the triplets (nl,n2,n3) for which AKJ/AK3 > 1. We know that all the triplets listed in Table I will be included because it was shown earlier that if AE1/AE3 > 1 it follows that AK1/AK3 > 1, but it is likely that additional triplets will be included in the sample for which AK1/AK3 > 1. (2.7) and (2.8) show that the condition is fulfilled if q < n2 + n - qA < (2.19) which is satisfied provided n2 < n* = (1/4 + A)1/2 - 1/2 (2.20) 24

24 23 - _ n',n3)= 22 -21 20 -19 / __ 18 17 16 - =15 14 13 n3 2 - 11 10 9 8 - 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 n1 Figure 1. The necessary condition for the position (nln3) is between the curves nI = n3 and F(nl,n3) = 0 in order to provide a possibility for satisfying the condition AEj/AE3 > 1. 25

TABLE I VALUES OF n2, FOR GIVEN VALUES OF nl AND n, FOR WHICH AE1/AE3 > 1 3 8 9 10 11 12 13 14 15 16 17 18 19 20 4 5 5 6 6 6 6 7 7 7 7 8 8 8 8 8 9 8 9 9 9 9 9,10 9,10 10 9 10 10 10 10 10,11 10,11 10,11 11 10 11 11 11 11,12 11,12 11,12 11,12 11,12 11 12 12 12 12,13 12,13 12,13 12,13 12 13 13 13 13,14 13,14 13,14 13 14 14 14 14,15 14,15 14 15 15 15 15,16 15 16 16 16 16 17 17 17 18 As before we must satisfy the selection rule n3 - nl < n2 < n3 (2.21) but it is straightforward to show that n** < n3 which indicates that AK1/AK > 1 if and only if n3 - n < n2 < n* 3 1 2 ** (2.22) The necessary condition restricting the choice of (nl,n3) is found, as in the previous case, by the inequality n - n < n (2.23) leading to 2G- + (n2- < G(nn3 ) = n3 - (4n1-1)n5 + (n1-3n ) < 0 (2.24)

which is straightforward to solve. Figure 2 shows the region in which G(nl,n3) < 0. We find as expected that this region is somewhat larger than the region in which F(nl,n3) < 0, see (2.17). As in the previous case we calculate from (2.22), with the definition of n** in (2.20), the lower and upper limits of n2 for each pair (nl,n3) satisfying G(nl,n3) < 0. The results are given in Table II where the entry of a single number indicates that n2 can have this value only. On the other hand, an entry of a pair indicates that n2 can take all values between the smaller and the larger number. For example, it is seen from Table II that if nl = 9 and n3 = 18, n2 may take the values 10, 11, 12, 13, 14 all of which will results in AK1/AK3 > 1. In order to summarize the present investigation we have prepared Figure 3. The circles show those triplets for which AK1/AK3 > 1. For each pair (nl,n3) we have listed the minimum and maximum values of n2 below the circle. Those circles for which AE1/AE3 > 1 have been filled, and the corresponding values of n2 are given above the circle. The open circles indicate the triplets for which AK1/AK3 > 1 and AE1/AE3 < 1. In the sample given on Figure 3, i.e., 1 < nI < 18, 1 < n3 < 20, there are 229 triplets for which AK1/AK3 > 1, but only 86 with AK1/AK3 > 1 and E1/AE3 > 1, giving 143 cases with AK/AK3 > 1 and AE /AE3 < 1. It is thus seen that the latter category contains the majority of the cases. 3. Concluding Remarks The main purpose of the present note has been to analyse a simple threecomponent system in a barotropic, non-divergent fluid with respect to energy and enstrophy transfers. Using the reasonable restriction that we consider active systems only we have found general conditions which determine the ratios of energy and enstrophy transfers to the large and the small scale for a given change on the intermediate scale. It cannot be concluded that there is always more energy transferred to the large scale and more enstrophy going to the small scale, but the investigations provide the formulas necessary to determine the ratios for any given three-component system. 4. Acknowledgments The research has been supported by the National Science Foundation under Grant No. GA-16166. Mr. James Pfaendtner programmed the calculations leading to the determination of the criteria. 27

40 I I 1 38 - ~38 ~G(nln3)=O 36 34 32 30 -28 -26 -24 /22 n3 n20 - 18 12 10 10 - Figure 2. The region between the curves n1 = n3 and G(nl,n3) = 0 gives the positions of the point (nl,n3) necessary, but not sufficient, to satisfy the condition AKI/AK3 > 1. 28

I r I I I I i I I I I I I I I I I 11,12 12, 1 13,14 14 5 15,16 16 17 18 20 0 0 0 o 0 0 14 13,15 12,15 11,15 12,16 13,16 14,16 15,17 16.17 17,18 18 19 11 11,12 12,13 13,14 14,15 15 16 17 19 - O o 0 * 0 0 * *0 0 0 0 14 13,14 12,14 11,14 11,15 12,15 13,1514,1615,16 16,17 17 18 10,11 11,12 12,13 13,14 14 15 16 18 - O0 0 0 ~ ~ ~ ~ ~ ~ 0 13 12,13 11,13 10,14 11,14 12,14 13,15 14,15 15,16 16 17 17- 00 0 4 4. 0 710 10,1111,12 12133 14 15 12 11,12 10,13 10,13'11,13 12,14 13,14 14,15 15 16 9,10 10,11 1112 12 13 14 16 o o ~ 0 0 * o 11,12 10,12 9,12 10,12 11;13 12,13 13,14 14 15 9 9,10 10 11 12 13 15 o- ~ ~ ~ ~ ~ ~ o0 11 10,11 9,11 9,12 10,12 11,12 12,13 13 14 8 9 10 11 12 14 0 0 0 0 0* ~ ~ 0 10 9,10 8,11 9,11 10,11 11.12 12 13 13 0o o 8 9 10 11 13 o o ~ ~ ~ ~ o 9 8,10 8,10 9,10 10,11 11 12 7 8 9 10 12- 0 ~ ~ 0 8,9 7,9 8,9 9,10 10 11 n3 7 8 9 8 7,8 7,8 8,9 9 10 6 7 8 O 10 o ~ o 7 6,7 7,8 8 9 6 7 9 - O 0 0~ 0 6 6,7 7 8 5 6 8 - O 0 5,6 5,6 6 7 7 0 0 0 5 5 6 6 o o 5 0 4 4 3 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 n1 Figure 3. The circles indicate the points (nl,n3) for which AK1/AK, > 1, if n2 has any one of the values listed below the circle. The dots indicate the points (nl,n3) for which AKJ/AK3 > 1 and AE1/AE3 > 1 if n2 attains any of the values common to the sets listed above and below the dot. 29

61 61 9T 91 9T LT (9T'LT) LT LT 9T (LT9'9) (LT'9T) 9T 9T T (L'TT) (9T1'T) (9T1'T) 5T T tT (9T1'T) (9T~'T) (6T'T) (6T'+T) 1T QT T (91'Qr) (Qr'Qr) (Q1'i[) (8T'Qr) (tr'Q[) ~1 (9TCT) (~TCT) (6T T) (QTCT) (~T'T) T CT FT (9T'FT) (T'FT) (tT'TI) (tT'IT) (T'FT) (T'FT) FT ZT TT ('T11T) (1T'TT) (iT'TT) (T1'T1) (lT'Tl) (ZT'TT) (ZT'TT) TT TT OT (1T'ZT) (IT'T) (7T'OT) (T'OT) (TI'OT) (ZT'OT) (T1'OT) (TT'OT) 01 0O 6 (5T1'T) (tI'TI) (~'11) (CT'T0) (T1'6) (Ti'6) (TT'6) (OT'6) (OT'6) 6 6 9 tT (7T'QT) (QT'aT) (Zl'TT) (ZT'OT) (TT'6) (TT'9) (01T9) (6'9) (6'9) 9 9 L T il lT (ZT'TT) (TT11'OT) (OT'6) (OT'9) (6'L) (8'L) (9'L) L L 9 TT OT 6 (6'9) ('L) (L'9) (L'9) 9 9 9 L 9 (9') (9'5) 5 t ~ ~ Z oz 6T 9T LT 9T 5T +T T T TT OT 6 L 9 8 Tu T < K:V/TIV HOIHM ~O0 Qu CEV Tu 0o SaImVA MnISO OeJ u To SmIVA 0 1^\ II arIavI

References Charney, J. G., 1966: Some remaining problems in numerical weather prediction, Advances in Numerical Weather Prediction, Traveler's Research Center, Hartford, Connecticut. Fjartoft, R., 1953: On the changes in the spectral distribution of kinetic energy for two-dimensional, nondivergent flow, Tellus, Vol. 5, pp. 225 -230. Platzman, G. W., 1960: The spectral form of the vorticity equation, J. Meteor., Vol. 17, pp. 635-644. Saltzman, B., 1970: Large-scale atmospheric energetics in the wave number domain, Reviews of Geophysics and Space Physics, Vol. 8, pp. 289-302. Saltzman, B. and S. Teweles, 1964: Further statistics on the exchange of kinetic energy between harmonic components of the atmospheric flow, Tellus, Vol. 16, pp. 432-435. Steinberg, H. L., A. Wiin-Nielsen and C.-H. Yang, 1971: On nonlinear cascades in large-scale atmospheric flow, J. Geophys. Res., Vol. 76, pp. 8629 -8640. Wiin-Nielsen, A., 1972: A study of power laws in the atmospheric kinetic energy spectrum using spherical harmonic functions, Meteorologiske Annaler, Vol. 6, No. 5, pp. 107-124. 31

A NOTE ON THE ANGULAR MOMENTUM BALANCE OF THE ATMOSPHERE by A. Wiin-Nielsen Department of Atmospheric and Oceanic Science The University of Michigan Abstract The vertically averaged meridional transport of momentum of the atmosphere is calculated under steady state conditions from a simple parameterization of the surface stress. The primary purpose of the calculation is to demonstrate that the major features of the emomentum transport as derived from atmospheric data can be reproduced by the simple model. 33

1. Introduction The qualitative aspects of the required meridional momentum transport of the atmosphere can easily be derived by a consideration of the surface stress which reduces the westerly momentum in the regions of surface westerlies and increase the westerly momentum (decrease the easterly momentum) in the region of surface easterlies. In order to maintain the momentum balance in a zonal ring extending from the surface of the earth to the top of the atmosphere it is necessary to transport westerly momentum from the the regions of surface easterlies to the regions of surface westerlies as first suggested by Jeffreys (1926) and later used by many authors. The fact that the surface stress may be calculated by a vertical integration of the equations of motion or, alternately, from the vorticity equation has been used to obtain information on the geographical distribution of the stress from upper wind information. Adopting a relation between the surface zonal stress and the surface zonal wind as is generally done in models of the large-scale atmospheric circulation it is equally possible to calculate the vertically averaged, meridional transport of momentum. It is thus possible to ask if it is possible to reproduce the momentum transport as derived from atmospheric winds from the observed distribution of the surface zonal winds. The calculation of the zonal stresses from calculated momentum transports has been carried out by several authors (see, for example, Wiin-Nielsen et al. (1964)). In the following sections we shall give a particularly simple example which may be used for educational purposes because it can be solved in a straightforward manner by analytical methods, but the general method may also be used to compute the meridional transport of momentum from a representative meridional profile of the zonally averaged wind. 2. Formulation of the Problem It is well-known that the zonally averaged form of the first equation of motion may be written in the form uz 1 X(uv)z cos2 cp (uu) z - acs+ + ~ fv + F (2.1) at a cos2 z, in which u, v, and co are the three components of the three-dimensional velocity vector, a is the radius of the earth, cp is latitude, p is pressure, and F.,z is the zonal average of the zonal component of the frictional force per unit 34

mass. The subscript z denotes a zonal average, defined by the relation f- 2 f21( ) d (2.2) The zonally averaged form of the continuity equation is 1 byVz coscp _ 7z acosp..+.- = 0 (2.3) a coscp -p dp Adopting the boundary conditions oZ = 0 at p = 0 and p = Po = 100 cb where the last condition is the common approximation and introducing a vertical average by the definition _ _ f ( ) dp (2.4) it follows from (2.3) that zM = (2.5) Using (2.5) and the boundary conditions stated above it follows by applying (2.4) to (2.1) that?z M 1 2 (uv)zM c0o2 p +-II- F (2.6) at a cos2cp c,zM( We shall now adopt the following expression for F F = - gp (2.7) where g is the acceleration of gravity and T the zonal component of the stress. It follows then from (2.4) that F = - - (2.8) FzM p XP,z"o where the subscript o denotes the value at p = p. We shall now further adopt the parameterization, common in large-scale models that 355

T p c V u (2.9),z,o 0o d o zo where po is the density, cd the surface drag coefficient, and Vo the windspeed at p = po. Introducing finally a steady state assumption in (2.6) we get 2 1 a M cos c g V 2- -- p c V u (2.10) a cos2cp d o zo where we have introduced the notation M = (uv)z,M. It is (2.10) which normally is the basis for a qualitative discussion of the requirements for the momentum transports in the atmosphere. (2.10) says that there must, under steady state conditions, be a convergence of the momentum transport when uz > 0 and a divergence when uz < 0. If we can get a numerical estimate of the coefficient to Uzo in (2.10), and if uzo is specified in some fashion we may use (2.10) to calculate M by integration of (2.10). 3. A Simple Example It is generally recognized that cd and VO are variables to which we have to assign typical values. We shall be using a normalized form of M when computed, and it is therefore of no great concern which numerical values we assign. Let us denote ~g _gcd Vo = p PO Cd V - RT (3.1) p o 0 Using g = 9.8 m sec2, cd 3 x 103, Vo = 10 m sec-1, R= 287m sec deg and To = 2880K, we find s = 3.56 x 10-6 sec-1 The meridional distribution of uzo, applicable to steady state conditions, is characterized by easterlies in the low latitudes, westerlies in the middle latitudes, and weak easterlies in the high latitudes. A simple expression which behaves in this fashion is uzo = - U cos 4p (3.2) We shall now calculate M from (2.10)using (3.2) for uzo. We obtain M cos p aU l/2 os(4p) coscP cp 3.3) cp = - aU cos(4p) cos p dcp 3.3) ocp 36

which upon integration gives M = A sin(2cp) (3-4sin cp) (3.4) where 2 -2 A = (1/6)EaU = 18.9 m sec (3.5) adopting Uo = 5 m sece Figure 1 shows M calculated from (3.4) and normalized in such a way that the maximum value is unity (left side). The same figure shows uzo (right side) normalized in the same way. A comparison between Figure 1 and similar figures based upon observed winds prepared by Starr et al. (1970), shows that the simple example depicts the major features of the "observed" distribution of M although it should be noted that the figures given by Starr et al. (loc. cit.), shows M cos2cp. This difference accounts for the much smaller values shown in the negative transports in the very high latitudes in the study based on observations. We shall next calculate M from a representative profile of uz(c). For this purpose we have calculated the profile from the climatological-data for the Southern Hemisphere givenbyVan Loon et al., (1971). The annual average of uzo, computed as the mean of the data given for the four months: January, April, July, and October, is shown in Figure 2. M was computed by a numerical integration of (2.10). The values of M, normalized in such a way that the minimum value is -1, are shown in Figure 3 as the solid curve. The agreement between the momentum transport from our simple example shown in Figure 1 and Figure 3 is obvious indicating that our example is quite representative. The circles entered on Figure 3 is the total momentum transport for the Southern Hemisphere calculated on the basis of Obasi's (1963) investigation which used IGY data for the calendar year 1958. The general shape of M from these data is the same, but the northward transport in the high latitudes is considerably larger in 1958 than in the climatological average. 4. Concluding Remarks The principle used in this note has been used several times to calculate the surface stress from upper wind statistics. We have emphasized that a parameterization of the surface stress in terms of the surface winds permits a 37

UzO -0-8 -0-4 0.0 0-4 0-8 / I- I II I I I / 80- _ 80 -70- 60 60-,50 50 50 '-3 5 0 ^Vl -o -4-M ''40 * O 30 40 \ -40 / 30- 30- / 20 10 10 2 0-0 0-2 0-4 0-6 0-8 1-0 M Figure 1. The right part shows the specified zonally averaged wind at the surface (upper scale) as a function of latitude, while the left part shows the meridional momentum transport, both normalized with respect to their maximum value (lower scale). -0-:

Latitude 00 0 0 0 0 0 0 0, Co \ / C) C) N 0 00 Figure 2. The zonally averaged surface geostrophic wind in m sec-1 as a function of latitude. The curve is the annual average for the Southern Hemisphere.

0 0 10 0 0 20 0 30 a) * _ -i -I 40 0 50 0 0 70 0 0 80 0 -1.0 -0.8 -0-6 -0-4 -0-2 0.0 0-2 0.4 0.6 M Figure 3. The meridional, vertically averaged momentum transport computed from the data given in Figure 2. Circles are obtained from Obasi (1965).

calculation of the total meridional transport of momentum. Such calculations, using a simple analytical example and statisitics of the surface wind, have been carried out with fair agreement with other direct calculations of the vertically averaged, meridional momentum transport. References Jeffreys, H., 1926: On the dynamics of the geostrophic winds, Quart. J. Roy. Meteorol. Soc., Vol. 152, pp. 85-104. Obasi, G.O.P., 1963: Poleward flux of atmospheric angular momentum in the southern hemisphere, J. Atmos. Sci., Vol. 20, pp. 516-528. Starr, V. P., J. P. Peixoto, and N. E. Gaut, 1970: Momentum and zonal kinetic energy balance of the atmosphere from five years of hemispheric data, Tellus, Vol. 22, pp. 251-274. Van Loon H., J. J. Taljaard, R. L. Jenne, and H. L. Cratcher, 1971: Climate of the upper air: southern hemisphere, Vol. II, zonal geostrophic winds National Center for Atmospheric Research, NCAR TN/STR-57. Wiin-Nielsen, A., J. A. Brown, and M. Drake, 1964: Further studies of energy exchange between the zonal flow and the eddies, Tellus, Vol. 15, pp. 261-279. 41

UNIVERSITY OF MICHIGAN 9015 03527 570