Dynamics of Atmospheres and Oceans, 4 (1980) 143--184 143 Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

BAROTROPIC AND BAROCLINIC INSTABILITY IN ROTATING STRATIFIED FLUIDS

PETER D. KILLWORTH

Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW (Great Britain)

(Received May 2, 1978; revised March 22, 1979; accepted September 12, 1979)

ABSTRACT

Killworth, P.D., 1980. Barotropic and baroclinic instability in rotating stratified fluids. Dyn. Atmos. Oceans, 4: 143--184.

The linear normal mode instabilities of a parallel shear flow which varies both vertically (z) and meridionally (y) in a quasigeostrophic, rotating, stratified fluid are considered. The ~ effect (variation of Coriolis parameter with y) is included. Both two-layer and con- tinuous fluids are treated. Attention is concentrated on the types of instability possible for a given shear flow. It is found that the instability can be described adequately by three nondimensional parameters: k, the ratio of the horizontal length scale of the shear to the internal deformation radius; ~, which is either the ratio of layer depths in the two-layer fluid or the fractional depth of variation of the stratification in the continuous fluid; and ~, suitably nondimensionalized.

Asymptotic analyses, confirmed by direct numerical solutions, are performed for condi- tions in, which various parameters become large or small. The ~ effect is essentially quanti- tative, whereas k and ~ define the type of instability as barotropic (if the kinetic energy of the mean flow feeds the growing perturbations), baroclinic (if the available potential energy of the mean flow feeds the perturbations) or mixed (a combination of the two).

The case of large k (the most relevant for oceanographic applications) is treated in detail. It is shown that y-independent problems have only limited relevance. For a fixed deformation radius and the y scale of the mean flow increasing without limit, the asymp- tote is not the case of no y variation in the mean flow.

1. INTRODUCTION

The topic of mean flow instability has existed as long as modem fluid

dynamics. The continued interest in the subject is largely a result of the many

geophysical phenomena which may or do depend on instabilities for their

production: atmospheric and oceanic eddies, vacillations in the laboratory

or atmosphere, and many others. Study of instabilities in this context has been essentially two-fold: in

homogeneous fluids, begun by Rayleigh (1880), with much later work by Lin (1955), Drazin and Howard (1966) and many others; and in nonhomo-

144

geneous fluids, begun by Charney (1947) and Eady (1949) and extended by many authors. These analyses were one-dimensional in character, whereas most real flows are two- or three-dimensional. Except for some general theo- rems, and some analytical work on slowly varying mean flows, most studies of two-dimensional mean flows have been numerical. Thus a unified treatment of the instability of a uni-directional mean flow remains lacking, and it is not easy to see how to provide one. However, a first approximation to such a theory would be useful.

The object of this paper is first to locate the parameters in any given situa- tion which control the possible types of instability. Analysis is then performed for asymptotic values of the parameters; this permits a specification, to within an unknown O(1) numerical factor, of the perturbation length scales, growth rates and type of instability (barotropic, baroclinic, or whatever), and an understanding of the mechanisms involved.

Both the two-layer approximation and continuous flow are examined, the former not because it is very realistic, but because of the frequency of its occurrence in the literature. Results for two-layer and continuous flows are often very different.

Sections 2--5 of the paper set up the problem and derive the three relevant parameters. Sections 6--9 explore the asymptotic ranges of these parameters; Section 10 discusses briefly how the ranges would be connected. Section 11 discusses the relevance of performing local or representative calculations, and includes some time-dependent solutions.

2. E Q U A T I O N S OF MOTION

It is assumed throughout that quasigeostrophic dynamics is applicable and that the fluid has a flat bot tom. The relevant equations (e.g., Charney and Stern, 1962; Pedlosky, 1964a) constitute an expression for conservation of potential vorticity. Examination will be made of small perturbations to a zonal flow, of the form (a) ui(y), i = 1, 2 for a two-layer fluid with mean depths H1 and H2, or (b) u(y, z) for a continuously stratified fluid. Aaces are taken with x eastward, y northward (for geophysical applications) and z mea- sured vertically upwards from the fluid surface; u represents the eastward velocity. A single Fourier mode with an x dependence of the form exp(ikx) will be assumed, where k is an east--west wavenumber.

The investigation will concentrate on normal modes of the system, propor- tional to exp(- - ikc t ) . If c (= c r + ici) has a positive imaginary part, the flow is unstable. The conservation equations for a two-layer fluid are

(Ul --C)[~/lyy--k2~l + ( 1 / a 2 ) ( ~ 2 - - ~ 1 ) ]

+ [~*+ (1/a2)(ul - - u 2 ) - - u lyy] ~1 = 0 (2.1)

(u2 - c ) [ ~ 2 ~ , - k 2 ~ 2 + ( ~ / a 2 ) ( ~ 1 - ~2)1

+ [~3" + (5/a~)(u2 -- u l ) - - u2yy] ~2 = 0 (2.2)

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where $i, i = 1, 2 represent stream functions for the perturbation to the upper and lower mean flows, /‘I* is the y derivative of the vertical component of the Coriolis vector f, 6 is the depth ratio H,/Hz, and a is the radius of deforma- tion (gHl)1’2f;1 ;g is the reduced gravity and f. the average value off.

For a continuously stratified fluid, the equivalent of (2.1), (2.2) is

(U - c)(ti,, - kzti + [(fo2W)hu) + (P* - [(f,2/mhi, - h,)ti = 0 (2.3)

where N(z) is the buoyancy frequency. This needs vertical boundary condi- tions that the vertical velocity vanish at top and bottom, or that

(u - c)rcI, = UZJI, 2=0,-H (2.4)

where H is the depth of the fluid. It will be assumed that the flow is unbounded horizontally (the effects of

vertical boundaries will be discussed in the conclusions). Furthermore,

Ui and u become independent of y as y + +a, ; (2.5)

in many cases, Ui and u tend to zero for large I y 1. The relevant boundary con- ditions on tii and \L are, from (2.5),

J/i, $ are bounded asy + km; (2.6)

in many cases this will involve decay at infinity. This condition excludes forcing from infinity (e.g., by a wavemaker).

Both two-layer and continuous equations define an eigenvalue problem for c; of particular interest is the value of the growth rate kci.

3. NONDIMENSIONALIZATION

It is convenient to nondimensionalize these equations to bring out the dependence upon certain parameters of the system. Let L be the horizontal scale of the mean flow and U its characteristic velocity. Then, putting

(x, y) = L(x’, y’), (u, c) = U(u’, c’), k = L-‘k’, /3* = Up/L2 (3.1)

one obtains for the two-layer case, dropping primes,

!v-cahyy - k2h + h2(92 - $111 + [P + X204, -342) - u~yyltcl~ = 0

(3.2)

(u2 - c)[ Jlzyy -k2$2 + ~X2($r - $211 + [P + 6h2(u2 - ~1) - ~2yylti2 = 0

(3.3)

These equations contain three nondimensional parameters which determine the characteristics of the solution. These are

A= L/a, (3.4)

which is the ratio of the horizontal length scale to the radius of deformation.

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,~2 can also be thought of as an internal Froude number (Hart, 1974) or as an inverse measure of static stability. Also,

5 = H 1 / H 2

is the ratio of layer depths; and

(3.5)

(3.6)

is the effect of varying the Coriolis parameter. ~ is scaled with L 2, so that in many cases of interest ~ and ~,2 change proportionally.

It is desirable to nondimensionalize the continuous case to yield three parameters whose function and interpretation are as similar as possible to those in the two-layer case. This is a little awkward, as the radius of deforma- tion a becomes the largest solution of the eigenvalue problem

Wzz + ( N 2 ( z ) / ~ a 2 ) w = 0; w ( O ) = w ( - - H ) = 0 (3.7)

whose solution depends on the magnitude and structure of N 2 (cf. Gill and Clarke, 1974). An estimate of a is, however, easy to find. Let No be a scale for N, and suppose that N differs significantly from zero over a depth scale ~H (this definition of 5 differs from that in (3.5)). It will be assumed, when 5 < 1, that N decays approximately exponentially below depths of the order of 5H.* In the laboratory, 5 would be unity; in the ocean, 5 would be fairly small. Then a can be estimated by

a = N O 5 H / f o (3.8)

and this will be taken to define a for the continuous case. Then, scaling z on the depth H, (2.3) becomes, nondimensionally,

(U - - C) (~yy - - k 2 ~ / + X 2 5 2 [ ( ~ z / g 2 ) ] z ) + ( [J - - X 2 5 2 [ ( u z / N 2 ) ] z - - Uyy)~/ = 0

(3.9)

where ~ retains its original definition, X = L / a retains its original meaning (eq. (3.4)) and 5 is now the fractional depth scale over which N varies {i.e. 0 < 6 ~< 1). Equation 2.4 becomes

(u - - c ) ~ = u ~ , z = 0 , - -1 (3.10)

4. ENERGETICS

It can be shown that (cf. Hart, 1974; Gill et al. 1974)

d E / d t = T K E + T A P E (4.1)

where E is the total (kinetic + available potential) energy of the perturbations

* In the ocean, at great depths N can decrease by two orders of magnitude from its value near the surface.

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and, for the two-layer equations,

TKE = 1 Re(ik f d y ( U l y $ 1 ~ y + 5-1u2y $25;y))

TAPE = --(~ 5 2) Re(ik f dy(Ul -- U z ) ~ ~ )

or, for the continuous equations,

TKE = 1 Re(ikff dy dz u y ~ ; )

TAPE = 12 Re(ik f f dy dz u z ~(~ /N2)~ : )

(4.2)

(4.3)

(4.4)

(4.5)

Re denotes the real part, and an asterisk the complex conjugate. Equations 4.2 to 4.5 are dimensional; in nondimensional versions a s is replaced by k - z and f02 by k2~ 2.

The terms TKE, TAPE are usually interpreted as transfers of energy from the mean flow to the perturbations through the kinetic energy of the mean flow and through its available potential energy respectively; that is, through the momentum transfer and eddy buoyancy fluxes of the perturbations. Be- cause TKE and TAPE give an estimate of which transfer process is important, they allow a straightforward characterization of each instability process as either barotropic or baroclinic (Hart, 1974). If TKE >> TAPE then the baro- clinicity plays little part in the instability, and the process is barotropic instab- ility; conversely, if TAPE >> TKE then the process is baroclinic instability. If TAPE ~ TKE the instability is mixed. A different characterization of the instability would be necessary in the last case if one of TAPE or TKE were negative; this is not considered here.

5. NECESSARY CONDITIONS FOR INSTABILITY

The most useful guides in deciding whether a given flow is unstable are the necessary conditions for instability. This section draws heavily on the work of Pedlosky (1964a). What follows is a list of the necessary conditions, which take the same form for dimensional and nondimensional variables. The im- portant quantity is the northward gradient of potential vorticity q, defined by

q l y = ~* + ( I / a 2 ) ( U l ~ U2) - - U l y y ( 5 . 1 )

q2y = fl* + (~ /G2)(U2 - - M1) - - U2yy {5.2)

for the two-layer case (nondimensionally, k -2 replaces a z and fl replaces fl*), o r

qy = ~* -- [ (f~ /NZ)uz]z -- Uyy (5.3)

for the continuous case (nondimensionally, k252 replaces fo2). Then the neces- sary Conditions for instability are that

(i) (a) two.layer: qly and qzy between them must attain negative and posi- tive values (Pedlosky 1964a) (5.4)

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This can be achieved, for instance, if qly > 0 and q2y < 0 as well as by one of the q~y changing sign;

(i) (b) continuous: qy must change sign somewhere (Pedlosky, 1964a) (5.5)

(ii) (a) two-layer: at least one of ulqly and u2q2y must be positive some- where (Pedlosky, 1964a) (5.6)

(ii) (b) continuous: Uqy must be positive somewhere (Pedlosky, 1964a, Charney, 1973) (5.7)

This latter theorem has been generalized by Blumen (1968) to allow for non- parallel flow, where the condit ion is that dq/d~ must be negative somewhere; for an east--west flow, this reduces to condition (ii) (b).

The surface and floor boundary conditions can allow instability in the con- tinuous case even if (i) (b) is not satisfied. The alternative conditions are that

(i) (c) the sign of qy be opposite to that of uz somewhere at the surface, or

(5.8)

(i) (d) the sign of qy be the same as that of uz somewhere at the bo t tom (Charney and Stern, 1962; Blumen, 1968) (5.9)

Application of these conditions frequently permits the immediate classifica- tion of a given flow as stable, and will always give areas of stability in (k, 6, fl) space. For example, it is obvious from (5.1), (5.2) that sufficiently large k/3L will make qly and q2y of one sign, thus violating condition (5.4). Hence two- layer flows are stable in this case. However, the same is not necessarily true for continuous flows, from (5.8), (5.9). In many cases, these stability condi- tions are sharp (e.g. Pedlosky, 1964b).

There are several theoretical bounds on the eigenvalue c which can be used. Some of these are restrictions on c, and others restrict the growth rate kci. Pedlosky (1964a) has derived the relevant versions of all the known bounds. However, these prove to be of little use as predictive tools, and their primary function is as a check on calculations, and to give indications of for- bidden areas in the complex c plane. Because of this, the various bounds are not detailed here, although they will be used implicitly at various times in the analysis.

6. THE CASE OF SMALL ~: HORIZONTAL LENGTH SCALES MUCH SMALLER THAN DEFORMATION RADIUS

For the next few sections the behaviour of eigensolutions to (3.2), (3.3) and (3.9) will be studied. Since the case of general k, 5 and ~ requires numer- ical solution, the limit as various parameters become large or small will be studied. Some cases can be solved immediately, by use of the necessary con-

1 4 9

ditions for instability. In each limit the same pattern will be followed: the two-layer case will be examined, followed by the cont inuous case. Results will be given both dimensionally and nondimensionally, and illustrated by exact numerical results.

The first case is that of small ~,: the mean flow varies over horizontal scales which are much smaller than the internal deformation radius {alternatively, the static stability of the mean flow is large). Under such circumstances it seems likely that the primary mechanism for instability will be barotropic, i.e. the horizontal shear of the mean flow provides its instability.

(a) Two- layer case

It is necessary to estimate the orders of magnitude of k, ~, ~ and the y scale for the solution. Small values of k, ~ and 5 are contained within the analysis; large values of k are trivially stable; large i~l is stable by {5.4); and large 5 is dealt with below. Further, it is easy to see that the y scale can only be of the order of unity. Then, if

~i = ¢ 0 i + ~2~1~ + ... . i = 1 , 2 (6.1)

C = C O + ~k2Cl + . . . ,

substi tution into (3.2), (3.3) yields, to zeroth order in ~2,

(/21 - - C o ) ( ~ O l y y - - k 2 ~ O 1 ) + ~ ] O l ( f l - - U l y y ) = 0 (6.2)

(u2- -c0) (¢o2yy --k2@02) + ~02(fl--U2yy) = 0 , (6.3)

which are two statements of the Rayleigh instability equation for a homogene- ous fluid, first for a velocity u l (y ) , then for u2(y). The equations are discon- nected, so that one layer may be unstable without involving the other. Since no quanti ty in (6.2), (6.3) involves ~, or 6, their solutions are also independent of h and 6 in the limit of small ~. This decoupling is well known in equatorial systems, where a can be quite large (cf. Charney, 1963).

If @01 is unstable, then Co is specified as the eigenvalue of (6.2). In general, this is not an eigenvalue for (6.3), so that ~o2 vanishes identically. However, at the next order in ~2, ~12 is the solution of an inhomogeneous equation, forced by terms of order 6 VJ01, and will not vanish.

Thus the normal modes of the system consist of an O(1) per turbat ion in one of the layers, and a weaker flow, of the order of ~.2, in the other layer. T K E is much larger than T A P E as hypothesized, so that the instability is barotropic. A typical solution to (3.2), (3.3) in this r~gime is shown in Fig. 1.

Specific values for the entries in Table I need specific shear flows: Pedlosky (1964a,b), Hart (1974) and Philander (1976) each discuss a particular profile in great detail, using a variety of numerical methods. Kuo (1973) gives an excellent account of the known properties of (6.2). It appears, for example, that ~3 stabilizes eastward flow and destabilizes westward flow, although no proof of this appears to exist.

150

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0 0

0 . . . . ,

-4 -3 -2 -I 0

Fig. 1. Solution (amplitude and phase) to the two-layer problem, with u I = sech2y, u 2 = 0, when ~. = La - I is small. Parameter values are ~ = 0.2, 6 = 0.1, ~ = 0. The solution issym- metric abou t y ffi 0. The so lu t ion is opt imal , i.e. it has the fastest g rowth rate: h ffi 0,925, cr = 0.447, ci = 0.166. Also shown, by the dashed line is the op t imal so lu t ion o f the Rayleigh p rob lem, for which h = 0.904, c r = 0.452, c i = 0.178. As predic ted , the curves for 41 and the Rayleigh so lu t ion are very close.

Evaluation of c 1, the O(X 2) correction to c o, in general requires numerical solutions. Im(cl ) can be of either sign, although it appears numerically that for the fastest growing solution, Im(cl ) is negative. If this result is true, then

= 0 is a local maximum for instability, in the sense that nonzero ~ gives a weaker growth rate than zero X. This further suggests the possibility of an area of stability for X < O(1) in some cases (when Im(c) is zero); Hart (1974) gives an example.

Although irrelevant geophysically (except in regions of large-scale oceanic overturning, such as the Gulf of Lions or the Antarctic), it is of interest to allow 5 to become large while X remains small. The proofs are tedious and are omit ted; only a brief description is given here. If it is assumed that 5~ 2 is O(1), the expansion (6.1) yields two possibilities. Either the top layer has a Rayleigh instability, or the b o t t o m layer satisfies a modified Rayleigh equa- tion similar to the one-layer barotropic equation with divergence (cf. (7.4}}. However, both instabilities remain barotropic; the ratio TKE/TAPE is smaller, of the order of X-2, but still large.

(b ) Continuous case

The cont inuous flow problem allows an extra degree of freedom in the type of solution possible: the vertical scale of the perturbation may be either unity or small. Each case is considered in turn.

If the z scale is unity, then the same remarks as to the size of h,/3 and the y variation hold as in (a). Then, making a similar expansion to (6,1), (3.9)

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becomes, to zeroth order,

(u --Co)(~0yy - -k2~o) + b3- -uyy)~o = 0, (6.4)

which is identical in form to the Rayleigh equation. However, u is now a function of z also, so that if Co is an eigenvalue at one value of z, it will not be at another; hence (6.4) has no solutions. An exception to this would be if u varies only slowly with z (i.e. u~ is O(X) or less). Thus, decoupling can- not occur in the continuous case, unlike the two-layer case.

The neglected vertical derivatives of @ must therefore be important some- where. This can take three forms: a boundary layer at surface or floor; a con- tinuously rapidly varying function of z; or a region at intermediate depth in which @ is nonzero, vanishing rapidly above and below.

Let us consider first the boundary layer, at the top for definiteness. The natural depth for this, from (3.9), would be ~. However, no solutions are pos- sible. To see this, we pose a stretched variable

~ = z ~ - : (6.5)

At distances O(k) from the surface, u and N are independent of ~', and axe denoted by an overbar. For large ~', 40 must vanish. Then (3.9) gives

([~--C0)[ ~/Oyy --k2~J0 + (62//%r2)1~/0~'~ "] + ( f i - ~yy)~0 = 0 (6.6)

and (3.11) gives

~0~ = 0, ~" = 0 (6.7)

provided that (u -- Co) is not (in some sense) small. Equation 6.6 is separable, as N is constant. However, (6.7) necessitates a Fourier cosine transformation

0 O(Y, a) = f C0(Y, ~') cos a~" d~" (6.8)

This solution cannot decay as ~ tends to ~ irrespective of whether a is real or imaginary. (Substitution of (6.8) into (6.6) shows that 4o is g ( y ) cos ao~" for some function g(y), and constant ao-)

Condition (6.7) precludes solutions with exponential decay in the vertical. This condition may be relaxed provided that u -- c is suitably small some- where at the surface, so that both terms in (3.11) may balance. In the y-inde- pendent case, discussed by Miles (1964) and McIntyre (1972), the Charney (1947) short-wave solutions would occur. These are confined (exponentially) to a narrow layer in the vertical, have large wavenumber k, and exist for all ranges of ;~. They are essentially baroclinic in nature.

Their growth rates, however, are very weak, In Appendix A such modes are discussed in the case of u varying with y *. It is shown that Charney short-

* The appendices to this paper may be obta ined from the author upon request; they are omit ted here due to their length.

153

wave modes can only occur in highly selected locations and under specific conditions: these modes are confined strongly in the y direction as well as vertically; and that their growth rates remain weak. These solutions cannot be obtained for two-layer fluids, as noted by Simmons and Hoskins (1976). Given the narrow scales, numerical solutions would need very fine horizontal and vertical resolution for such modes to be found.

The other possibilities are that WKBJ-like solutions of the form

~(y, z) exp(-+D(z)k - 1 ) (6.9)

may exist, which oscillate rapidly in the vertical; here ~, D vary on depth scales of the order of unity. For a given (unknown) c, D,(z; c) would be ob- tained by substituting into (3.9) and solving the resulting horizontal Rayleigh- like equation with Dz as eigenvalue.

The nature of such possible solutions depends on the number and nature of the turning points of D,(z), i.e. the values of (complex) z where Dz(z) vanishes (where the WKBJ theory fails, and use must be made of connection theorems across the turning points). Since solving (3.9) is arduous numerically, the nature of the turning points cannot be determined in general. The qualitative description which follows (the inverse case of large k, discussed in Section 8) must therefore be regarded as essentially unproven, save for some numerical confirmation discussed below.

If there are situations where a single representation of the form (6.9) is valid for --1 < z ~< 0, then c must satisfy a quantization condition of the form

0

f D~(z; c) dz = n~ki, n = 1, 2, 3, (6.10) o o @

- -1

to satisfy the boundary condit ions (3.10), and this determines the eigen- value c.

When there are two or more turning points, a single representation (6.9) is not normally valid over the whole depth range, and the solution becomes more complicated, involving rapid decay with z. Another quantization condi- tion becomes relevant instead, and the solution then takes the form of an ex- ponential decay both above and below a region of (approximate) oscillatory behaviour with depth. (In general, this region is at mid-depth; solutions trap- ped at z = 0 or --1 do not normally occur.)

In the latter case, the usual form of solution is the lowest vertical mode (higher modes, i.e. those with more oscillation in the vertical, tend to be stable). Such a solution can then also be obtained by expanding the problem in powers of ~1/2 about some (complex) value z0, with the vertical scale of the order of k1/2 also. To lowest order, the 'local' Rayleigh problem, for u = ~(y) - - u(y, z0), must be solved, Co being determined as a function of z0. To the next higher order, it turns ou t that dc/dz must vanish at z0, which deter- mines z o. In general, c o is not typical of any 'local' Rayleigh problem unless u, vanishes somewhere for all y, when z0 is real. To a higher order still, the

154

zA I Y y,

r Y2 Y3 z : 0

.°.

Fig. 2. Schemat ic of con t inuous so lu t ions when )~ = L a - 1 is small. Full lines show [ ~l as a f u n c t i o n of dep th for various y ; dashed lines show Ph(~) . If u z = 0 at some dep th , then Ph(~) varies on O(~ I/2) scales also.

variation of ~ with depth is found as a parabolic cylinder function, of the form exp[ - -a (z --Zo)2)~ - 1 ] for some complex constant a, Re(a) > 0.

The solution is thus confined in the vertical, over a depth range O(X 1/2), with rapid oscillations (in phase) on depth scales O(X) (because z0 is com- plex). A schematic of this solution is given in Fig. 2. The instability is baro- tropic.

Pfister (1977) numerically solved a continuously stratified atmospheric problem for small X, and confirmed the above description; his profile had uz identically zero at mid-depth, so that Zo was real; thus the phase in his solu- tion varied on O(X 1/2) depth scales, as well as the amplitude.

Hence there are important qualitative differences between the two-layer and continuous systems when X is small; the main difference is the appearance of small vertical length scales in a cont inuous fluid, which cannot occur in the two-layer version.

7. THE CASE OF SMALL 6 : STRATIFICATION CONFINED TO A SMALL FRACTION OF THE DEPTH

This section considers the effects of making the upper layer very thin (or, equivalently, the lower layer very thick). The internal radius of deformat ion is of the same order as L, so that k is of the order of unity. (In physical terms, then, N would have increased as 6 decreased.) ~ may be considered to be of the order of unity; there are no qualitative changes if it is large, save the pos- sibility of stability.

In the continuous case, small 5 means that N differs significantly from zero * only in a thin layer near the surface (or floor), with a rapid decay on a height scale of the order of 6 below (or above) that. It is not assumed at this stage that u necessarily has the same behaviour with depth as N.

* In reali ty, a value much smaller than its near-surface value.

155

(a) Two-layer case

The y scale and k may be taken to be of the order of uni ty without loss of generality. Expanding

~i = ~0i + 8 ~ 1 i + i f 1, 2 .... (7.1)

C = C O + 5C 1 + .. . ,

(3.3) yields, to zeroth order,

(U2 - - C o ) ( ~ 0 2 y y - - ]~2 ~ 0 2 ) + (~ - - U2yy) ~02 ---- 0 (7.2)

which is the Rayleigh problem for the lower layer, as in Section 6. The upper layer satisfies

(U 1 __ Co) [~ /01yy __ (~2 + ~2 ) ~01 ] + [ ~ _ _ U l y y + k 2 ( U l __ U 2 ) ] ~ 0 1 __--

--k2(Ul -- Co)~o2 (7.3)

Therefore, two solutions are possible. The first is that the bot tom layer is barotropically unstable, and drives the top layer through (7.3). Both ~bol and ~bo2 are of the order of unity, and all quantities are independent of 6. The second solution has ~bo2 zero (i.e. ~2 is of the order of 5, forced by ~bol) and ~bOl satisfies a modified one-layer barotropic equation with divergence

(U 1 - - C0)[I]J01yy - - (k 2 + k 2 ) l ~ o l ] + [~ - - U l y y -t- ~,2(u I - - u 2 ) ] 1~/01 -- O (7.4)

Thus the instability requirements are the same as those in Section 6: for small 8, instability in the two-layer case requires that (5.4), (5.6) are totally satisfied by one layer alone. Again, both instabilities are barotropic. A typical solution to (3.2), (3.3) in this r~gime is shown in Fig. 3.

It is possible to examine the change in Ira(c) with 5 by evaluating Cl. This appears numerically to be positive for the fastest growing mode (again, no proof of this exists), indicating that zero 5 is a growth rate minimum in (~, k, 5) space.

It can be shown that as k increases (but with 5k 2 remaining small) only the first of the above solutions remains; (7.4) becomes stable, even if the condi- tions for instability in the upper layer are met (except when u2 = constant). Hence for moderate ly large k, only lower-layer Rayleigh instability is possible.

In the special case of no flow in the bot tom layer, it is possible to extend the above analysis considerably. The details are given in Appendix B, in which an expression is derived for the way the eigenvalue c varies with k and fl when 5 is small:

ca = 2 kcc~ (7.5)

so that c may be found at all points of the (k, ~) plane if it is known at ~ = 0 or k = 0, for example. For large k (with 5k 2 still small) it is also shown that

c ~ (constant -- ~)/k2 (7.6)

156

, ,¢

~V

2~

~2

d

Fig. 3. S o l u t i o n s to t h e t w o - l a y e r p r o b l e m , w i t h u 1 = s e c h 2 y , u 2 = 3 t a n h y s ech ~'y ( p l o t t e d in Fig. 5) w h e n 6 is sma l l . P a r a m e t e r va l ue s are ~ = 1, ~ = 0 .1 , 13 -- 0. T h e s o l u t i o n is o p t i m a l , w i th k = 0 . 5 7 9 , c r = 0 . 7 7 9 , c i = 0 . 1 1 9 . T h e c u r v e for 4 2 is v e r y c lo se to t h e o p t i m a l s o l u t i o n o f t h e R a y l e i g h e q u a t i o n w i t h u = u2 , as i n d i c a t e d b y t h e d a s h e d l ine, for w h i c h k = 0 . 5 8 7 , c r = 0 . 7 8 4 , ci = 0 . 1 1 6 .

(which satisfies (7.5) exactly). The form of the solution is independent of k.

(b ) Cont inuous case

(i) u varies vertically over the entire depth If k is assumed to be O(1) (for large k and small 5, see Section 9) then the

behaviour of (3.9) as 5 -* 0 depends crucially on the vertical structure of u. The region where N is of the order of 1 is of height ~ f u varies on vertical scales of the order of the overall depth then (u~/N2)~ is large below depths of the order of 5. Then ~ is forced to have an O(1) depth variation also, and the balance of terms becomes, for depths larger than O(6),

(u -- c)( ~ / N 2 ) ~ ~- (u~/N2)z ~ (7.7)

This has the solution

~(y, z) = A(y)(u -- c) = ~(y, 0)(u -- c) / (~(y) -- c) (7.8)

where ~(y) again represents the value of u(y, 0). Since for depths smaller than 0 (5 ) the vertical derivatives are 0 (5 ) and thus small, (7,8} subst i tuted into (3.9) gives, near the surface,

( ~ - - c ) [ ~ ( y , 0)yy - - k 2 ~ (y, 0)] + (13-- ~yy)~(y , 0) = 0 (7.9)

Therefore, (7.8) is a solution provided that the surface velocity is unstable in the Rayleigh sense. (In the intermediate region, where 8 2 (1/N 2 )z is O(1), (7.8) still satisfies (3.9) because both (7.7) and (7.9) are satisfied.) This solu- tion exists, in fact, even when k > > 1. The instability is baroclinic.

157

The other possibility is that the disturbance is confined near the surface, in a layer of thickness 5. If ~" = z5 -1 is defined as a stretched coordinate, then to leading order (3.9) becomes, in this layer,

(~ - - c ) [ 4 y ~ - - k 2 ~ + X2(4~/N2)~l + ( ~ - - ~ y y ) 4 = 0 (7.10)

with boundary condit ions

4~- = O, ~" = O; 4 ~ O, f ~ ---¢¢ (7.11)

This system possesses separable solutions in y and ~', of the form 4 = a(y)g(~), if g satisfies

(g~ /N'2)~ = --a2g (7.12)

for some constant ~, together with (7.11). There are a countable infinity of such solutions, since N -~ 0 as ~ -- ---=. Subst i tut ion into (7.10) shows that a(y) satisfies the Rayleigh equation with k 2 replaced by k 2 + k2~ 2. Again, instability requires the surface velocity to be unstable in the barotropic sense. For moderate X, or ~, there is stability * because of the high-wavenumber cut- off to the Rayleigh problem (cf. Drazin and Howard, 1966). Despite strong vertical variation, the instability remains barotropic.

(ii) u varies vertically over a region o f depth 5 If now u z is O(1) only in a region of depth of the order of 6 (i.e. there is

only a barotropic velocity at depth) the solution depends on the behaviour at depth of 52X2(uz/N2), in (3.9). This will be large if uz decays with depth more slowly than N 2, or it may be small.

If 82X2(uz/N2)z is large at depth, either it is balanced by X252(4,/N2)z as in (7.7) (yielding the solution (7.8) again, although the instability can be barotropic or baroclinic depending on the structure of uz) or 4 must vanish at depth. In the latter case, (3.9) yields

(U - - C ) [ 4 y y - - k 2 4 + ) k 2 ( 4 ~ / N 2 ) ~ ] + [~ - - Uyy - - k2(u~/N2)¢] 4 = 0 (7.13)

in the surface layer z <~ O(5), where ~" = z5 -1 remains a stretched vertical coordinate, with boundary condit ions

(u - c) 4~ = u~ 4, f = 0 (7.14)

4 - - ' 0 , ~" ~ - - ,~

This is a fully two-dimensional eigenvalue problem for 4, and yields a mixed instability.

If 5sX2(u~/N2)z is small at depth, then 4~ must also vanish at depth so that (4~/hcz)~ does not dominate (3.9). Then, below 0(5) , (3.9) becomes

(a -- c)(¢yy -- k2¢) + 03--f~yy)~ = 0 (7.15)

* The Charney short-wave mode remains, of course.

158

where ~(y) and a (y) are the values of if, u for depths larger than O(5). For z < 0(5) , (3.9) gives (7.13) again, but with the second condition in (7.14) replaced by

-* ¢(y) , ~"-*--~ (7.16)

This again yields two solutions. The first is that ~(y) (i.e. the deep flow) is Rayleigh-unstable and drives the surface layer via (7.13), (7.16). This is baro- tropically~ unstable, and is only possible if ~ varies with y. The second solu- tion has ~ zero, so that the problem reduces to the two-dimensional surface problem (7.13), (7.14) as before. The equivalent of the mixed instability in the two-layer case was barotropic. This is a qualitative flaw in the two-layer model.

8. THE CASE OF LARGE k: HORIZONTAL LENGTH SCALES MUCH LARGER THAN DEFORMATION RADIUS

This section examines the effects of making the horizontal length scale L of the mean flow much larger than the radius of deformation a. This is the most relevant case oceanographically and, to a lesser extent, for atmospheric motions. Few authors have examined this parameter range in any detail. McIntyre (1970) examined the effects on the Eady problem of introducing a small, slowly varying additional mean flow. His main results, however, were derived for jet-like profiles, i.e. profiles for which, at some Y0, uy = 0, y.= Y0, for all z. (No sense of narrowness or strength should be implied by the term 'jet'.) Jet-like profiles are a rather special case, as will be shown. Stone (1969), Simmons (1974), Gent (1974, 1975) and Kim (1978) used analytical and numerical techniques for this parameter range, but all calculations were for jet-like profiles. Holland and Haidvogel (1980) give a detailed numerical study of a particular jet-like profile.

In addition to y scales of the order of unity, it is now possible for solutions to be confined horizontally, with small y scales. Indeed, this is the most relevant case since the growth rates are larger than for wider y scales. Each case will be treated in turn, the first only briefly.

(i) Solutions with y scales o f the order o f unity

If 6 and k are assumed to be of the order of unity (k of the order of k is trivially stable) and

= ~o)~ 2 + ~1, (S.1)

where one or both of Go, ~1 is zero (this allows ~ to vary as k 2 or remain con- stant), then an expansion of the form

= ~o + ( l / k 2)~1 + ... (8.2) c = Co + (1/)~2)c, + ...

159

is possible. Both two-layer and cont inuous systems have the same behaviour; only the continuous case is treated here. To leading order, there is stability unless ~o is zero (i.e. ~ is, effectively, very small or zero for all normal pur- poses). Provided that Go vanishes,

~o = A(y) (u -- Co) (8.3)

where A(y) , Co are determined from the next order in the expansion, which gives

(~PAy)y -- k2~PA + fJlAQ = 0 (8.4)

where

0

, (y, co) = f (u-co) --1

(8.5) 0

Q(y, CO) = f (U--Co) dz --1

Equation 8.4 has similarities to one studied by Pedlosky (1964a); its stability criteria are uninformative, but it is found numerically that (8.4), (8.5) do possess (baroclinically) unstable solutions. Their growth rates are of the order of unity, however, and are thus much weaker than those in subsection (ii) below. As k increases, these instabilities blend smoothly into those in (ii).

(ii) Solut ions with small y scales

An appropriate length scale for the perturbation is not obvious. Clearly (3.2), (3.3) or (3.9) suggests a length scale of ~-1 , i.e. the deformation radius, and this is usually the case. There are no solutions confined to regions within O(~.-') of some value of Y(Yo, say} and which vary solely on ~-1 length scales. This is because an expansion about such a point would yield equations of the form

Cyy = ~X2~, (8.6)

where a is a constant and ¢ a linear function of ~. Such equations have no solutions decaying as )q Yl becomes O(1).

The relevant length scale turns out (formally) to be X -'/2 *. The following local expansion method both generalizes Simmons' (1974) solution and puts it into a firmer footing. It cannot derive all solutions, and in that case a WKBJ method (in Appendix C) is necessary.

* A ~--2/3 length scale looks possible; this yields Airy funct ion solutions of complex argu- ment , which grow exponent ia l ly with y and are therefore not permit ted.

160

(a) T w o - l a y e r case

Let us choose, at the moment freely, a value of y (yo, say) and let 77 be a stretched y coordinate

= 'kt/2 (Y --Yo) (8.7)

Without loss of generality, k is 0(,~), so that

k 2 = ~2k2 (8 .8 )

Expanding ui about Yo by

ui = Uoi + ( v l ~ l l 2 ) u l i + ~ (~2 /~ )u2 i + ..., (8.9)

where uni is the nth derivative of u~ at Yo, and expanding 4i and c in power series in k -1/2, the O(k 2) terms in (3.2), (3.3) become

(Uol - C o ) ( - t ~ o 2 4 o l + 402 - 401 ) + (Go + Uol - U o 2 ) 4 o l = 0 ( 8 . 1 0 )

(U02 --Co)[ - k 2 402 + 5(401 -- 402)] + [GO + 6(U02 --U01)] ~]02 = 0 (8.11)

These are equations for the local problem

ul = ul(yo), us - u2 (Yo) (8.12)

in which neither shear nor solution varies with y, since the coefficients in (8.10), {8.11) are constant. Hence 4ol, 402 are proportional:

4 0 2 = - [ G o - (Uo2 - Co) - k ~ ( U o l - C o ) ] ( U o l - Co) - 1 4 o l

-= ~ 4 o l , say (8.13)

and Co satisfies the quadratic (Gill et al. 1974)

[ - ( 1 +/~o2)(Uol - Co) + ~o + u o l - Uo2] [ - ( a + t~oZ)(Uo2 - Co) + Go

+ 5(Uo2 -- Uol)] = 5(Uol -- Co)(Uo2 -- Co) (8.14)

To this order Yo is arbitrary and so, therefore, is Co. The O(k 312 ) terms give

(Uol -co)(-t~411 + 412 - 4 1 1 ) + (~o + u o l - u o 2 ) 4 1 1

+ ( ~ u l l - cl)(--/~o2 4 o l + 4o2 - 4 o l ) + ~ ( u l l - u 1 2 ) 4 o l = 0 ( 8 . 1 5 )

( S o 2 - Co) [ - t~ 2 412 + 5 ( 4 1 1 - 4 1 ~ ) ] + [~o + 5 ( U o 2 - U o l ) ] 4 1 2 +

+ (~?u12 --Cl)[--ko24o2 + 5 ( 4 o l - 4o~)] + 5~(u12 --u11)4o2 = 0 (8.16)

Because of (8.10), (8.11), the 41i terms disappear after cross-multiplying to give, after use of (8.13),

4o1(A77 + B c l ) = 0 (8.17)

where A is linear in the uli and, for a given Yo, A and B are known. This can only be satisfied if

A = cl = 0 (8.18)

161

in general. The first of these determines Yo (and therefore Co) as a function of k o, usually uniquely. The value of Yo is, from (8.18), precisely the point at which

(dc/dy)lyfy 0 = 0, (8.19)

where c is a solution of the local problem (8.12). In other words, at Yo both real and imaginary parts of c have stationary values. Thus Yo is not in general the point where the local vertical shear is a maximum; neither is it the point where the local growth rate is a maximum. Under normal circumstances Yo is complex. In Gent's (1974, 1975) and Simmons' (1974) cases Yo was real, in fact the origin. This was because their jet-like profiles, possessing a point where uy vanished, automatically satisfied (8.19). Hence in general the solu- tion is confined to a narrow region around a predetermined complex value o f y .

The O()~) terms of the expansion give the variation of ~oi with rt:

(u01 - Co) (¢Olnn - k02 ¢21 + ¢ 2 2 - ¢ 2 1 ) + '7u11( - /~2 ¢11 + ¢ 1 2 - ¢ 1 1 )

+ [ ( 1 ~ 2 / 2 ) U 2 1 - - C 2 ] ( - - h O 2 ~J01 + ~ 0 2 - - ~J01) + ~JOl (7~2 /2 ) (U21 - - U22)

+ ¢111~(Ull -- //12) + ¢21(~0 + U01 -- 1/02) = 0 (8.20)

(//02 - -C0)[~J02r l~ --]~02~J22 + ~(~/21 - - ~/22)] + 7~U12[--/~2l~/12 + ~(t~/11 - - ~/12)]

+ [(172/2)U22 - - C2] [--]~02~02 + ~(~J01 - - ~ 0 2 ) ] + ~ ~J02(772/2)(/Z22 - - U21)

+ ~127~(U12--UlI) + ~22[~0 + ~(U02 -- UOI)] = 0 (8.21)

in which the Ujyy terms of qjy do not yet appear, so that the available poten- tial energy of the mean flow is the only source of energy for the perturbations. Cross-multiplying eliminates the ~2i but not ~1i, giving an equation of the form

E~01~ + F@02~ + G'~@II + H'7~12 + K,72~01 + L,72@02 + MC2~ol

+ Nc2~Jo2 = 0 (8.22)

where E, F .... N are known constants. From (8.15), (8.16), ~JlI, ~J12 are of the form

@il = ~P@ol + ~6(~); ~J12 = ~/~(~) (8.23)

where P is a known constant, and/~ is irrelevant since it cancels in (8.22). Then, using (8.12), (8.22) becomes

~/o1,1n = (QC2 + R~2)~Jol (8.24)

where Q, R axe known constants. Solutions with @Ol bounded as I~/I becomes large m a y be found only when

c2 = [ - - 2 R 1 / 2 ( n - - 1 / 2 ) ] / Q (8.25)

162

for some integer n > 0, when

~ o l = exp(--R1/2772 / 2 ) H , ( R 1 / 4 ~ ) , (8.26)

where H , is the nth Hermite polynomial and Re(R 1/2) > 0 defines the branch of the square root. The integer n is the number of antinodes of the solution *

If I m ( R I / 2 / Q ) is positive (apparently always so in jet profiles, when the ~1i terms do not enter; Simmons, 1974) then the correction term c2 acts to decrease Im(c), so that the fastest growing mode has n = 1, and

c = Co - - R I / 2 / ~ Q + O(~ -3/2) (8.27)

and

~01 = exp[--kRl/2(Y -- Y0)2/2] (8.28)

As noted, the jet profiles of Simmons (1974) and Gent (1974, 1975) were special in that Y0 was real. This implies that the eigenfunctions decay and oscillate with a meridional (y) scale of )~-1/2, or dimensionally (La) 1/2, as found by Simmons (1974). When Y0 is complex, however, this is no longer true. Then, on the real y axis, ~ol still decays away from a real point {not Re(yo), but normally close to it because Im(y0) is fairly small in practice) with a scale of ~,-1/2, but oscillates both N--S and E--W on the scale )~-1 (i.e. the deformation radius a). This confirms the original hypothesis of Stone (1969) regarding the y scale of perturbations, although his own calcula- tions, dealing with jet profiles, would normally possess the >,-1/2 variation only. These length scales are not in agreement with those predicted by Orlanski and Cox (1973).

An example of this behaviour is shown in Fig. 4, for the n = 1 mode, for the mean flow (shown in Fig. 5)

ul = sech2y, u2 = 3 tanh y sech2y, 5 = 0.1 (8.29)

chosen so that the shears in both layers are similar in magnitude, and no obvi- ous symmetries exist. The maximum shear (ul -- u2) is 1.88, at y = --0.52. Figure 4 shows that the phase varies linearly near the maximum of l~l; this is not near the point of maximum vertical shear. The phase changes slowly (although at a rate O(~.)) with y because Im(y0) is small. In higher modes phases change very rapidly due to the polynomial behaviour of the Hermite functions.

Three special cases can occur. The first is when the shear in one of the layers (e.g, the second) is weak, i.e. lu2yl < O()~-1/~), which can happen if u 2 is small, or varies much more slowly than ul. Vanishing A requires u l y to be zero, or Y0 to be real and at the point of maximum vertical shear: the profile is thus jet-like.

The second case is when R vanishes in (8.24); for a jet profile this involves

* The algebra here also applies to Section 6, if ~--1 replaces )% and y replaces z.

iq, ill 05 I

0 ='

163

Ph~ x )-3 -z d~ . , _ _ ; L . . , . . _ 0 I

Fig. 4. Optimal solution for the shear in Fig. 5 with ~ = 8, ~ = 0, ~ = 0.1. Other param- eters are k ffi 4.278 (k 0 = 0.536), c r = --0.666, c i ffi 0.287. Although the phase changes slowly with y near the maximum, its variation is nonetheless O(k); higher modes show very rapid changes in phase due to the Hermite polynomial behaviour. Similarly, for larger ~ the width of the perturbation decreases as k -1/2.

Ulyy, U2yy vanishing at the jet. Simmons (1974) shows that the relevant length scale becomes ~-~/¢n+2), where n is the first power of T} in further expressions like (8.24) to have a nonzero coefficient. Except for jet profiles, however, the oscillatory (eddy) scale remains ;~-I.

The last special case is that of a turning point at infinity, where all deriva- tives of ui vanish. Analysis shows that solutions bounded at, say, +oo and decaying at --~ are vanishingly small for values of y less than O (log ,k), with a growth rate corresponding to the local value at +~. The same applies for profiles in which all derivatives vanish at a finite value of y; the case con- sidered by Hart (1974) is such a special case, although his boundary condi- tions precluded growth in this manner. Figure 6 shows a typical result for the

Ul u2

Fig. 5. The profile u I = sech2y, u 2 = 3 tanh y sech2y, used in several numerical examples, and chosen to possess no symmetries.

164

t i !

i¢! i 4

0 5 I

J i

I

r P

3 ~

ol -2 0 2 y z.

Fig . 6. S o l u t i o n f o r t h e s h e a r u 1 = t a n h y, u 2 = 0, w i t h ~ = 5, ~ = 0, 5 = 0 . 1 . O t h e r p a r a m - e t e r s a r e k = 2 . 5 (k 0 = 0 . 5 ) , c r = 0 . 1 6 8 , c i ffi 0 . 2 1 2 . A l l y d e r i v a t i v e s w e r e s e t t o z e r o a t y = co. T h e v a l u e o f c is e x t r e m e l y c l o s e t o t h e l o c a l v a l u e a t +co, n a m e l y c r = 0 . 1 6 7 ,

c i = 0 . 2 1 5 .

profile (u~ = tanh y, u 2 = 0); the vertical shear at infinity permits baroclinic growth there.

Let us suppose that I m ( R 1 / 2 / Q ) is negative in {8.23). Then Im(c) is increased by the correction term c2, and by choosing n large, Im(c) can appar- ently be made arbitrarily large. This does not invalidate the analysis, but does indicate that alternative methods of description are likely to be necessary.

A suitable method is based on the WKBJ formulation (Stone, 1969, Gent, 1974, 1975), which permits evaluation of all the eigenvalues but gives less information about the form of the solution. The method poses

~i = ~i(Y) exp(kD(y)) (8.30)

which allows for rapid y variation through the XD(y) term. Details of the solution are given in Appendix C. It is shown that Dy (y) is given by the quadratic

[(u 1 --c)(D2y-- k 2 - 1) + Go + ul -- u2l[(u2 -- c)(D2y - - k 2 - 6)

+ ~0 + 5(u2 -- ul)] = 6(ul - - c ) ( u 2 - - c ) (8.31)

and that c is determined from the quantization condit ion (along the anti- Stokes line; see Appendix C)

~2

f Dy dy = (n - - l /2) iTr/X, n = 1 , 2 , 3 .. . . (8.32)

165

where ~1, ~2 are the two (complex) values o f y at which Dy, a s a function of y and c, vanishes. The integer n corresponds precisely to that in {8.25). For small n, ~1 and ~2 are nearly equal; when they coalesce, at y = Yo, say, then dc/dy vanishes at Yo, so that the point of coalescence is also the po!nt about which the local expansion is made. This specifies c and Yo.

Solutions are then found by altering c by an amount O(~ -1 ) such that (8.32) is satisfied. For small n, the solutions are formally equivalent to those found by the local expansion; for larger n, c moves further away from coales- cence on a path in c space.

An example of this is shown in Fig. 7, for/3 o zero, and for ko varying from zero to (48)1/4 = 0.79, the high-wavenumber cut-off for the y- independent problem. Since ~o is zero, the quadratic (8.31) factorizes. The paths in c space for any given ko all have Im(c) decreasing monotonical ly (i.e. the first mode is the fastest growing) and all tend towards the origin as the value of the integral tends to infinity in modulus. The diagram has been confirmed by numerical solution of (3.2), (3.3), and the errors in c are O(k-3/2).

In many ways Fig. 7 is the most " typica l" curve, at least for zero ~o. For given k o and ~ there are an infinite number of solutions, dense at the origin. All but the first few of these have Im(c) < < 1, so only the first few modes would be observed in practice. None of the solutions coincides with a local solution at any real value of y (save possibly for profiles with nonzero shear at infinity), a topic discussed further in Section 11.

Nonzero ~o changes the situation drastically, as Fig. 8 shows. Let us con- sider first positive ~o. This, corresponding to eastward flow, appears to stabilize the flow (no proof of this familiar result exists to the author 's knowledge). Numerical results, as indicated, are again O{~, -3/2) from the

\ %

:', -o:8 o -0'6 -0-4 -0.2 Cr

0.5

0.4 ci

0.3

0-2

04

Fig. 7. The locus of the p o i n t of coa lescence in the c o m p l e x c p lane , for the shea r in Fig. 5, for k 0 varying f r o m 0 to 0.79. O t h e r p a r a m e t e r s are ~ = 0, ~ = 0.1. In add i t i on , the pa th in c space fo l lowed by c, as t he m o d u l u s of the integral (8 .32) increases, is s h o w n , for k0 .= 0, 0 .48 and 0.73. The first two curves are e x t e n d e d to a value of (8 .32 ) corre- s p o n d i n g to the first m o d e for ~ ffi ~ /10 . T he lowes t curve is e x t e n d e d to s h o w h o w the origin in c space is app roached . A so lu t i on of (3 .2) , (3 .3) for )~ ffi 6 .91 lies at the end of the a r row; its beg inn ing shows the c -pa th e s t ima te . The er ror is O(~--3/2). T h e r e are c pa ths emerg ing f rom real coa lescences for k 0 > 0 .79 ( n o t shown) .

166

A

i3o= o-~ 6o:-O~

"z *: 3~ 0,~' %: ~8 \ oc e C~r~ x C ~ ~ t o G e 0 ~= 68 ~' \ \ 02

".:,:o= ~9 -~5 -I -0-~

Cr

Fig. 8. The locus o£ the point of coalescence in the complex c plane, for the shear in Fig. 5, varying kO, and ~0 = +-0.1, 6 = 0.1. In addition, various paths in c space are shown for different values of/{0 ; the endpoints of the curves are chosen only for clarity, and have no significance, except for/{0 = 0.48, ~0 = --0.1, which could not be traced further numerically. For ~0 = +0.1, the two end-points of the coalescence curve (on the real axis) do not mark the ends of the 'acceptable' coalescence, for smaller values of/~0 (e.g. 0.28, as shown) the coalescence is real, but the c path complex. For k 0 ~> 0.73, the c path could only be traced a short distance; this is confirmed by results in Fig. 151 For ~0 = --0.1, one end-point of the coalescence curve is at k 0 = 0:38, beyond which the curve still exists (dashed line) but is unacceptable. The dashed curve continues beyond k 0 = 0.28, but is not shown. Solutions of (3.2), (3.3), for ~. = 8, lie at the end of the arrows; their beginning shows the c path estimate. Both errors are O(k-312). Finally, for ~0 = --0.1, the paths of solutions to (3.2), (3.3) for k 0 = 0.28, are marked by dotted lines as ~t increases from 8 to 11. As predicted, these do not lead asymptotically to the 0.28 coalescence.

predicted solution, and all c paths eventually tend to the origin. A high-wave- number cut-off (ko = 0.73) exists. There is a low-wavenumber cut-off for coalescence (ko = 0.33), although for any finite distance along the c path there is instability for much smaller ko, as shown by the ko = 0,28 curve. For • ~o ~< 0.39, the path is one of increasing Im(c). This means that for large k, the fastest growing mode is not the first, but a much higher one (i.e. the local analysis is valid, but Im(c2) is positive).

When ~0 is negative (westward flow) there is destabilization, as the right- hand diagram in Fig. 8 shows. A high-wavenumber cut-off (0.83) exists, and for k o > 0.48 the situation is essentially unchanged (numerical solutions are again indicated). However, for k o ~< 0.48 trouble was experienced in tracing the c path too far numerically because of the branch cut problems in (8.32). The continuation of the c path, although not found numerically, is indicated schematically and confirmed by the first- and second-mode solutions found by direct solution of (3.2), (3.3).

At k o ~ 0.38 the situation changes abruptly. The coalescence still exists, indicated by the dashed line, but there are no solutions close to coalescence. (The details are complicated, and are omit ted here.) Hence the permissible c path ends abruptly, far from coalescence, with a nonzero value for the integral, i.e.

~2 dy f Dy /> T (8.33)

167

for some constant T. As k -* ¢¢, low-numbered modes cannot exist, as {8.33), {8.32) imply that n must be O(X) > > 1.

As an example, no first-mode solution for the profile (8.29) with ho = 0.28, ~3o = --0.1 exists, confirmed by direct solution of the equations. Second- mode solutions for ho = 0.38, 0.28 and 0.18 were all found for k = 8, and the approximate c path followed to k = 11 by direct solution for ho = 0.28 as shown. However, the ho = 0.28 c path is not tending towards the 0.28 coales- cence, which is as predicted.

The situation is thus very complicated. However, the general rule applies throughout: the fastest growing mode has wavenumber, growth rate and scale of y variation all of the order of k. Usually, but not always, the fastest growing mode has n = 1, and is confined to a width of the order of k -1/2.

(b ) Con t inuous case Although (3.9) is not separable in y and z, one can pose a WKBJ-type solu-

tion, as in the two-layer case, of the form ~ = ~(y, z) exp(kD(y)) . To leading order D~ is determined as an eigenvalue of the (complex) local y- independent vertical problem, if it is assumed that c o is known. Lacking knowledge of how these eigenvalues vary with y, Co or k (which would involve numerical solu- tions of problems similar to those of Charney, 1947) one cannot proceed analytically, although there seems little doub t that the analysis would be similar to the two-layer case. This is confirmed, for Eady-type solutions, by Gent (1974, 1975). The local expansion technique remains tractable, how- ever, as follows.

Expanding again about some Yo in powers of ~ - i /2 and retaining previous notation, (3.9) gives, to O(~2),

/~0(~0)---- (U0(Z)--C0)(--k02~O + ~2(~, /N2)z) + (Go--52(Uoz/Y2)z)~o =0

(8.34)

where u0(z ) is the first term in the expansion of u(y, z) about Y0. This is the local y- independent solution at Yo, with boundary condit ions

~ 0 8 ( ~ o ) - (Uo -- Co) ~oz -- Uo, ~o = 0, z = 0, --1, (8.35)

and is fully discussed by Green (1960), Charney and Stern (1962) and Gill et al. (1974). This gives ~0 in the form hO?)/g(z), where g is determined and h is not. To next order,

~ 0 ( ~ i ) + ~ i ( ~ o ) = 0 (8.36)

where ~?l is a linear operator with coefficients proport ional to 77 and c i , involving ui(z ). ~ i satisfies boundary condit ions also of the general form (8.3.6). Eliminating ~ i between (8.35), (8.36) (involving cross-multiplying and integration from top to bo t tom) gives

A~7 + Bc i = 0 (8.37)

168

where A, B are known constants. As in the two-layer case,

A = cl = 0 (8.38)

so that again

dc0/dy = 0, y = Yo, (8.39)

which thus defines Yo and Co. Finding Y0 would he more difficult than in the two-layer case, as it involves the iterative solution of a complex two-point boundary value problem.

Equation 8.36 then gives an expression for 41. To next order, (3.9) gives, with obvious notation,

• ~0(~12) + - ~ l ( ~ J l ) + (/-t 0 --Co)~]OrTq + .~2 (~ /0 ) "~ 0 , (8.40)

where ~22 involves r/2, Uyy and c2. Eliminating ~2 gives, after some algebra,

h ~ = (Qc2 + RT? 2) h (8.41)

where Q, R are known constants. From here the theory is identical with the two-layer theory, as are the conclusions about length scales of perturbations, etc.

The foregoing analysis gives no physical clue why short horizontal length scales are selected in the y direction. Pedlosky's (1975) analysis provides the clue. He analysed the stability of the fastest growing mode to a y-independent situation; in this case the solution must also be y-independent. However, his analysis showed that these perturbations were themselves unstable, with meri- dional scales of the order of X -1 in addition to the already existing x scale of X -1. Thus, when there is variation in the y direction (forced by horizontal variations in shear) then the solution automatically breaks down into small y scales and the solutions discussed here emerge.

9. THE CASE OF L A R G E ~ AND SMALL 5 : H O R I Z O N T A L L E N G T H SCALES MUCH L A R G E R T H A N D E F O R M A T I O N RADIUS, AND S T R A T I F I C A T I O N C O N F I N E D TO A SMALL F R A C T I O N OF T H E DEPTH

This asymptotic analysis combines features of the previous two sections, and is included because of its oceanographic relevance, Many details will be omitted for brevity. The precise degree of smallness of ~ as a power of X will usually be taken as

5X 2 = 50 = O(1) (9.1)

(a) Two . layer case

As in the case of large X, two possibilities exist: when the y scale is of the order of unity, and when it is small. Solutions with 0(1) length scales are considered first.

If u 2 is not constant, an expansion in powers of 51/2 (i.e. X - i ) shows that

169

no instability exists unless 13 is O(1) (i.e. ~o = 0) and k = O(1). A little algebra then shows that, to leading order, 42 satisfies the Rayleigh equation in the bo t tom layer, and hence instability requires the lower layer to be Rayleigh- unstable.

If u2 is now taken to be constant or zero (a special case) then the Rayleigh instability disappears and is replaced by a weaker mode. Several types are possible, depending on the sizes of k and ~, bu t in all cases the growth rate is O(6) < < 1, and is thus weak. The instability is mixed. For 5 smaller than k - 2 (k - 4 , say) there is a barotropic instability with a possible O(1) growth rate; if u2 is constant, this becomes an O(k -2 ) growth rate.

The other possibility is that the y scale of the perturbation is small. The y-independent (local) problem has been considered for small 5 by Gill et al. (1974). They showed that, if~o is nonzero, c~ ~ 51/2 for small 6, 42 is O(51/2), and k0 is selected to within O(51/4). If ~0 is zero, then ko is O(61/4) rather than O(1), and c~ is O(51/4) also. Furthermore, the sign of ~o must be oppo- site to that of ul -- u2, otherwise there is no instability.

Many of their results apply to when u varies with y. If 5 is small, but 5k 2 is still large, then the turning points and coalescences of Section 8 are close to the real axis; c~ is O(51/2), and k 0 is selected exactly as if there were no y variation. The growth rate is thus 0(51/2k) which is approximately constant on lines of constant 5k 2 in (k, 5) parameter space.

Clearly this growth rate ceases to be O(k) when 5k 2 -~ O(1). At this stage, problems appear in the asymptot ic methods, so these results were examined numerically, allowing 5 to vary with various powers of k. In one set of calcula- tions, 5k 2 was held fixed while k increased. For vah ,~ of k up to 11, c i ~ 51/2 ~ k -1 . Larger values could not be achieveu due to rounding errors. In a second set k was held fixed and 5 decreased. Initially, while 8k 2 >_ 0.1, c~ varied as 51/2; as 5 decreased further, however, ci varied as 5 and there were indications that the decrease was becoming faster than the change in 5. This is because the next order correction to c,. begins to dominate.

Thus cl ~ 511~- is a useful rule of thumb provided that (a) ,k is not too large and (b) 5 is not too small. Beyond these values, the instability is much weaker; these asymptotics have not been explored.

(b ) Con t inuous case

There are again two subcategories, depending on the behaviour of u with depth as 8 becomes small.

(i) u varies vertically over the entire depth As N -* 0 at depth, it is assumed again that k252(uz /N2)z is very large at

depth (as before, there are other possibilities). Thus either 4 vanishes at depth, or

(u - - c ) ( 4 z / N 2 ) z ~- (u z /N2) , 4, i.e. 4 -- A(y)(u -- c) (9.2)

170

exactly as in (7.7), the size of X being irrelevant. The solution (7.8), {7.9) still applies, provided that the surface u velocity is Rayleigh-unstable (i.e./30 = 0, among other things). The instability is baroclinic; X, 6 do not now enter the problem. If instead ~ vanishes at depth, a stretched vertical coordinate ~" = z6 -1 yields, for ~" O(1 ),

( U - C)(l~yy - - k 2 ~ + ~,2(@r/N2)f ) + [ f i - /~yy --60Uz(y , 0)(1/N2)f] ~ = 0

(9.3)

which is of the form (7.10), but with an extra term, and it is easy to see that this is stable, except for Charney modes, by WKBJ methods.

(ii) u varies vert ical ly on d e p t h scales 0 ( 5 ) Clearly ~z must vanish at depth, since otherwise there is nothing to balance

the ~z terms in (3.9). There are two possibilities. If ~ becomes independent of z below 0(6) , then a Rayleigh equation of the form of (7.15) is produced at depth, where all terms in (X, 6 ) have disappeared, together with the bound- ary value problem (7.13), in which X still occurs. One possibility is thus that there is Rayleigh instability at depth. This involves fi0 vanishing, and k, ci are of the order of unity and independent of (X, 6). The instability is baroclinic.

Alternatively, ~ vanishes at depth, and (7.13) becomes an eigenvalue prob- lem for large X. In this case the form of the solution is identical with that in Section 8. The only difference is in the lower boundary condition, applying at ~" -* - -~ rather than z = --1. The solution is confined to a narrow (X -1/2 ) y scale, with phase variation on scales O(X-1). k is O(X), ci 4. O(1) and all quantities are independent of 6. This is qualitatively different from the two- layer case, which can possess no solution of this form; in particular, there is no tendency for growth rate to vary as 6 1/2 for small 5. Again, the instability is baroclinic.

10. TRANSITION REGIONS: PARAMETERS OF THE ORDER OF UNITY

The foregoing sections have a t tempted a complete definition of instability types in various relevant asymptotic parameter ranges. When all of ~, 6 and are of the order of unity, these analyses fail. A theoretical discussion of this final volume of parameter space is beyond the scope of the present study. Instead, this section wilt examine the behaviour of the solutions already derived, as one of the parameters passes through the transition region where that parameter is of the order of unity. If solutions can be identified on each side of the region, this indicates that no additional (non-asymptotic) solutions have been omitted. We can, furthermore, expect instabilities to be mixed for all parameters of the order of unity, as indicated by Section 4.

The role of ~3 is predominantly quantitative, as has been shown. In other words, let us suppose that, for a given mean flow, ~ is small and there is some instability present. If ~ is increased through unity, it begins to affect qy (cf.

171

Section 5). If the flow is two-layer, then qly and q2y become one-signed for large Ifll and the flow is stabilized. If the flow is continuous then the distur- bance becomes confined to the surface or floor as Ifll increases, and there is (steadily weaker) instability for larger and larger k; eventually only the Charney short-wave mode survives.

Letting 6 pass through unity has no meaning for a continuous fluid, and merely inverts the roles of the upper and lower layers for a two-layer fluid; hence changes in 6 can be ignored here.

The remaining case, of h passing through O(1), is of most interest. In a continuous fluid there is a smooth transition of instability types as ~. varies. When X is small the barotropic instability has O(1) horizontal length scales and a vertical scale O(X). As ~ increases towards unity this latter scale becomes O(1), and the instability mixed. As ~ continues to increase, the hor- izontal scales begin to contract and become O(7~ -~ ); the instability becomes baroclinic. Although no numerical solutions are presented to confirm this, it seems plausible that this is a complete description.

In a two-layer fluid it is possible to examine the problem in more detail.

C,.Ci - - Cr o.2s ~. " - -

- . . Cr

o [

0.15

kq

0 1

OO5

0

TAP /

ol 0-2

k

to: , " " " , I 0

0:6 ~ ;~ L- , :g

Fig. 9. Opt imal so lu t ion for the profile (u 1 = sech2y, u 2 = 0, fl = 0) as ~ varies. The top c u r v e shows Cr, ci, the middle curve kci and k, and the b o t t o m curve T K E / ( T K E + TAPE) . There is a change o f m o d e s at ~ ffi 1.01. Both modes con t inue to exist as subsidiary maxi- ma in g rowth rate for some way, as shown by the dashed lines. Note h o w nar row is the region o f mixed instabil i ty. The near-coincidence of c r, ci is for tu i tous .

172

Figure 9 shows how the instability with maximum growth changes as k increases from 0.2 to 1.8 for the f low (u 1 = sech2y, u 2 = 0,/3 - 0, 5 = 0.1}. For small X the instability is of Rayleigh type and is barotropic. It remains so as X increases to about 0.9 (when, for small k, there is mixed instability). At this stage there are two maxima in growth rate as a function of k. As X con- tinues to increase, their relative magnitudes change, and at X = 1,01 the fastest mode changes to a mixed instability, at smaller h. For X > 1.2, there is again a unique maximum in growth rate as a function of k (the relevant k now varies proportionally to k). For X > 1.4, the instability is baroclinic, with a narrowing y scale.

Two points stand out. First, the region of mixed instability is very narrow in X space (approximately 1 ~< X < 1.4); on either side of this region the high and low X asymptotic curves work remarkably well. This seems to hold for all profiles studied. Secondly, l~11 changes weakly with X across X ~ O(1) (although Ph( ~1 ) and I~21 vary strongly). Whether this result is true in general is unknown.

Thus there is again a smooth transition as X changes. The barotropic mode for small X becomes a weakly growing mode in the baroclinic rdgime (in the

log k " , B C , k - O(X} ,

M , k - O U } . .

Lu--O{1), st ~b[e i = " .

.,,J3- X 2 01$0 weOkly - " ~ unstoble 8C orM,"

" " - w-O(k " ' BT.W-O(), " " . LU- O(1) stobleif "" "-~ 3- X 2 " , ,

ST,k- O(1).W- OC}

BT = bo.rotrop;¢ M = m;xed

BC = baroct in ic

B =,k- O(X},uJ-O{X)

. . M , k ~ O ( 1 ) , W ~ O ( 1 )

x

\\\ BT.w- 0111,0}- 0(1)

\

\

' , s),2~ \

L ~

0 log 6

2 ~ - - _ ~. - - - - 0 " ~

b ' ~ Z s r 6 " ~-OIt

• -C (1 ' ~ O " ) .~'-0{ '

cJ BZ cr B" W-OC)

, - O { ' ~ - C ' " t.,~C!"

I D log8

Fig. I0. The (X, 8 ) diagram for the two-layer case, showing the nature of the fastest instab- ility (assuming that there is instability) and the wavenumber k and growth rate co. For zero /3, the region of large %, and small 5 is modified slightly.

Fig. 11. The (%`, 8) diagram for the continuous case. Case (a) refers tou profiles varying over similar depth scales to N, case (b) to u profiles varying over the full depth. No numer- ical solutions have been attempted for this case, so the conclusions are necessarily some- what speculative.

173

sense that kc~ ~ O(1)). Conversely, the baroclinic mode for high k changes smoothly into a low& barotropic mode for small k.

The results of the last few sections may be summarized as a (X, 8 ) plot, fol lowing Hart (1974) . Figures 10 and 11 show such plots for two-layer and continuous flows. Not all the features discussed previously can be included in a single diagram, since no one set of condit ions can cover all possible cases. The asymptotic regions analysed have been connected by transition regions to make the behaviour of the solutions as smooth as possible. There may, of course, still be extra structure in such regions which this analysis cannot dis- cover. Hart's (1974) calculations fit Fig. 10 well.

Computer examples of the opt imum growth rate, for a profile with mean flow only in the top layer, and the more general (8.29) are given in Figs. 12 and 13. These are in no sense 'typical' solutions, since many different profiles would be needed to cover the variety of special cases, areas of stability (such as Hart's 1974 calculations possessed), etc. However, all the main ingredients of the theoretical picture are included. The areas of mixed instability are again very narrow.

5

2

0 .5

0.15 I I -

2

I

0-5

5

////

.o.15

0 1 o

02o.o2 o:05 o:: o'.2 c'.5 o.z , , • o.o2 olo5 o.~ o'.2 o.s 5 6

Fig. 12. The (k, ~) diagram for the two-layer f low u 1 = sech2y, u 2 = 0, with ~ --- 0, show- ing contours of the fastest growth rate. The hatched area marks the approximate region of mixed instability; above it is the baroclinic region, below it the barotropic region. In the baroclinic region, growth rate is approximately constant on lines of constant 5)k 2, as predicted.

Fig. 13. The (k, 6) diagram for the two-layer f low in Fig. 5, with ~ = 0. The (baroclinic) growth rate is only constant on lines of constant 6~ 2 for rather small 6.

174

11. DISCUSSION: LOCAL C A L C U L A T I O N S

Table I presents estimates of the orders of magnitude of the parameters ~, 5 and ~ for various types of mean flow, in regions where quasigeostrophy could be expected to hold. In most cases, especially the open ocean, k 2 is large, and 5k 2 takes values either O(1) or large. ~ is usually of the order of k2 except for regions of strong boundary currents.

The fact that k is large in many geophysical contexts (especially the ocean) has made investigation of nonlinear mesoscale phenomena an expensive numer ical problem, since the resolution required is very fine. Many authors have sought to simplify the problem in some way. The most popular is to restrict both the geophysical area (to a zonal mean flow) and the dynamics (by using linear perturbations to the mean flow). Both in the atmosphere and the ocean (e.g. Haidvogel and Holland, 1978) this gives surprisingly good esti- mates of eddy sizes, growth rates, etc.

The relevance of such calculations is limited in two ways. The first is in the neglect of nonlinearity in the dynamics. The second is that the calculations are usually local, in that they omit any horizontal variation in shear. A mean flow u(z), and stratification, are assumed typical of the majority of the area under study, and solution of the local vertical eigenvalue problem is per- formed *. It is to this limitation that the present work is addressed.

Justification for local calculations varies. Sometimes (Robinson and Mc- Williams, 1974; Hart and KiUworth, 1976) the representativeness of the local solution as the first term in a WKBJ-type expansion is claimed explicitly; sometimes (Gill et al., 1974) the assumptions are more implicit. However, in nearly all cases there is a basic feeling in the literature that slowly varying (i.e. L > > a) mean flows can be treated locally, the variation in shear having only a weak effect. It has been assumed that as the length scale of the mean flow becomes large compared with the radius of deformat ion (L > > a), the local solution is the limit of the actual solution to the (now weakly) y-varying problem.

However, this is not the case, in general, as shown by Section 8, for many reasons. Except in the case of a jet (where Simmons (1974) and Gent (1974) state correctly that the actual solution tends to the local solution for large k), the actual solution has phase variations on the scale of k -1 , or dimensionally the deformation radius. These persist no matter how large k is, so that the actual solution does not tend to the local solution for large ,~.

Further, the expressions for c resulting from local solutions have no con- nection with the actual solutions to the problem with y variation. This is probably shown most graphically by Fig. 14, which shows the local solution to the problem with the mean flow (8.29) for various y, with the full range of ko (0 to 0.79) given. For a given finite k, the solutions are given by follow-

* This is less t rue for the a t m o s p h e r e t han for the ocean , in which on ly very few calcula- t ions (e.g. Pedlosky, 1 9 6 4 b ; Haidvogel and Hol land, 1978) were two-d imens iona l .

175

-0.6 -0

° 0 ~ £ -2

NC~, 2:,, , ,

~ ' , -',-1.~ ', ~ . ', ~, , , -.,.6 ',

, yj '. ,., , : ' , : ;

-I -0.5

~C~

"0-5

0,, ~' 0 Z 1.2 1-0 0.8

" 1 8 1 .6 1.4 ".. ," ," ,," . 0 . 6 • • / s " 2 ",." ,." ," "4' : O-~.,,,. ,'

0 0.5 Cr 1

Fig. 14. The firm curve s h o w s the coa l e scence curve for the profi le in Fig. 5, as in Fig. 8, for 0 ~ k 0 ~ 0.79. The dashed l ines s h o w the e igenvalues (c r, ci) for the local , y - indepen- dent so lut ion , at values o f the shear corresponding to the value of y indicated on the dia- gram, again for k 0 running from 0 to 0.79.

ing the c paths, as in Section 8. For fixed k, the solutions are a trail of points in c space, corresponding to the various modes, usually tending asymptotic- ally to the origin. With the exception of local solutions "near" coalescence, the majority of the local solutions have no relevance to the actual solutions.

This is plausible: a point which is locally stable would obviously give no guide to the actual instability problem, but it might be expected that some subset of the local solutions (e.g. the point(s) with maximum local growth rate) would give a good estimate for the actual problem. This is not the case, necessarily, as Fig. 15 shows. Local solutions are stable for k0 < 0 .32 or k 0 > 0.74, yet actual solutions exist for k well outside the ranges. In addition, the optimal local solution for k0 = 0.33, say, underestimates the actual growth rate by a factor of 6; the local solution also predicts incorrectly the value of k which makes the growth rate fastest, with an error of 15%. Nor is the location of the solution correct: the disturbance is maximized around y ~ --0.7, whereas the local growth rate is maximal at y = --0.52.

It might nonetheless be argued that a 15% error is quite acceptable for such a calculation. However, situations can occur in which local solutions give

0.3

koC ~

0"2

/ ~ o p t l m " J I c.¢t~aL solution ~ - ~ - ' ~ ~ 'Q. local solutlOn

0 0.T 0.2 0.3 0-~ 0.5 0.6 0.7 0.8 ko

Fig. 15. Growth rate k0c i against kO, for ~ = 0.1, ~ = 0.1, and the shear in Fig. 5. The top curve shows the local y-independent curve for YO -- 4 . 5 4 (the value of YO where the growth rate is largest); the lower curve is the actual solution to (3.2), (3.3) for ~k -- 8. k O = 0.79 is not a cut-off for the actual solution due to c paths emerging from real coalescences.

176

qualitative errors. Let us consider the mean flow given by (~ = 0, 5 = ().1) and

= e sech2(y -- 1), ~ = sech2y (11.1)

ul = (~ + ~ ) / ( 1 + ~), u2 = (~ --5~)/(1 + 5)

which represents a variable barotropic (~) addition far from a mean shear (= ul -- u2). For e = 0, this flow is almost identical to that displayed in

Fig. 12. Now the local stability properties and local solutions to (11.1) depend only on fi, and are independent of e. Thus all local problems are unstable, with maximal growth rates, k, c; independent of e (and cr merely has an addi- tion proportional to ~ ).

The actual solution to (11.1) looks quite different, however. Figure 16 shows the fastest growth rate, as e is varied. For e > 1.3, the problem is stable, although all local problems are unstable. This can be derived, in fact, by the methods of Section 8. Hence no local solution gives any indication of the stability, a qualitative error.

Of course, this analysis has only examined normal modes, which are not usually complete. Conventional wisdom, supported by analysis like that of Pedlosky (1964c), has it that a disturbance in a t ime-dependent calculation needs about one e-folding time of the fastest growing normal mode to orga- nize itself into the (vertical) shape of the fastest growing mode, and behaves from that time on as if it were that fastest mode. When y variation is added, one might expect that a disturbance might grow at a rate determined locally (i.e. by local vertical energy transfers) at least until nonlocal effects reach the area, presumably from the region where the y dependent normal mode has appreciable values.

Unfortunately, none of this is necessarily the case, ~ can be seen from a collection of initial value problems, integrated numerically. The equations satisfied are those for two layers, but with explicit time dependence. In Fig. 17, the mean flow is sech2y in the upper layer. The fastest growing nor- mal mode is strongly peaked about the origin, with a growth rate of 0.70.

o l koC; ,09

• 06 \

-04

O 0 .2 .c

\

.6 ,8 ~ ~.2 ~ -

Fig. 16. Optimal growth rate for the profile u I -- u 2 = sech2y, ~u I + u 2 = ~ sech2(y -- I), = 0, ~ = 0.1, for varying e. The local instability calculation gives only the mean vertical

shear (u I -- u2) , and gives instability at all y independent of e. For large e, however, the flow becomes stable. (At e ~ 0.52, there is a mode change.)

177

, :

(a) ,;.p:,:l

/ , , " / , ~ ' o)

/

. . . . . ~(-1.s)

/ / . .

I (~2 /

~5~o 2 ~ 6 ~ t lo ~'~

10 z

10

kpd

(b) / ~(o)

. .,~. / ~ ' "~

~6~o ~ ~ & ~ (o ,'~ t

102~ (c) Iq~,l / ¢¢0)

/ ,o

I . - / . / /

/ W! { ;. & ~ 1'o ,'£

t

Fig. t7. (a) Results of a time-dependent calculation with u I = sech2y, u 2 = 0, )~ = 8, k 0 = 0.5, ~ = 0, 6 = 0.1. The boundary conditions at infinity were replaced by ~/1 = ~2 = 0, y = +-4 (which in the y-independent case reduces all growth rates by less than 1/2%). The initial perturbation is ~1 = 0.1 sech 2 2 (y + 1.5), ~2 = 0, centred about y = --1.5. l~zl is shown as a function of time t at the origin and at --1.5, by the firm line. The local, y- independent growth rate (o~) at the origin is 0.86, and at y = --1.5 is 0.16. If the solutions were to grow at precisely these rates, they would follow, or be parallel to, the two dotted lines. The third dotted line shows the actual growth rate, as calculated from theory, of 0.70. The long dashed line shows similar results with the differing initial condition ~1 = ~2 = 0.1 sech 2 2(y + 1.5). The short dashed line shows a calculation with the original initial conditions, but with u I replaced everywhere by its value at y = --1.5, i.e. sech 2- (--1.5). (b) As (a), but ~1 = 0.1 cos(Try/8), ~2 = 0 at t = 0. (c) As (a), but ~1 --" 0.1 cos- (17/ry/8), ~2 = 0 at t = 0. The local growth rate at y = --1 is 0.36, as marked.

178

In Fig. 17a, a disturbance is initiated in the upper layer, and peaks strongly around y = --1.5 (a point suitably far from the origin). At this point, the local y-dependent growth rate is 0.16; for comparison, that at the origin is 0.86.

The time dependence of this example appears to bear out the above heuristic comments. Within times of the order of unity (an e-folding time for the actual solution) the value of @1 at the origin is growing at the rate of 0.70 (it does not attain 0.86, however). Simultaneously, ¢ at --1.5 decays slightly until t ~ 2, then increases asymptotically to a growth rate of 0.16, the local value. For times longer than those shown, when @(0)/@(--1.5) achieves its asymptot ic value, @(--1.5) of course grows at a rate of 0.70 like the rest of the solution. Thus, apparently, the local growth rate has dominated the nor- mal mode solution for times of the order of 10.

This conclusion, however, depends crucially on several factors. The first is the vertical shape of the initial disturbance. The solution with @1 = @2 ini- tially shows a growth rate at the origin much faster than its local rate, only settling down to 0.70 after times of the order of 5. Similarly, @(--1.5) grows at rates in excess of 0.20, 25% larger than its local rate. These differences, of course, come about through differing vertical energy exchanges. The sec- ond factor is the existence of horizontal shear. The solution (also shown in Fig. 17a) when u is replaced everywhere by sech 2 (1.5) bears this out. The growth rate becomes 0.12 to 0.13 at all points, which is the correct value for the system as a whole. Thus the increase of 30% on this value, achieved locally when u does vary with y, is at tr ibutable to nonlocal effects.

The third, and most important, factor is the initial horizontal shape of the disturbances. Figure 17a shows the behaviour when the disturbance is con- fined to a small region well away from the vicinity of the (usually unknown) normal mode solution. Figures 17b and 17c show the effects of varying the horizontal shape of @1. In Fig. 17b, @1 is spread smoothly over the entire (--4,4) range of y. In some sense, this distribution contains a great propor- tion of the fastest normal mode at the origin (where the local growth rate of 0.86 is never achieved), and, similarly, a great proport ion of the local solu- tion at y = --1.5. (Were the modes complete, this s tatement could be made quantitative.) For long times, the influence of the normal mode reaches y = --1.5, causing a sudden increase in the growth rate at that point.

If @1 varies rapidly with y, however, (Fig. 17c), then the behaviour away from the origin is quite different, consisting of weak oscillations followed after t ~ 9 by a rapid increase (as the solutions approach asymptotical ly the normal mode). There is no evidence of any local growth behaviour. Again it can be seen that the influence of the normal mode takes a finite time to reach y = --1.5.

The final example, Fig. 18, is of when the local problems are nearly all stable. The solution at y - --0.7, --1.5 grows rapidly with t ime (after an ini- tial decay at --1.5), tending asymptotically to a growth rate of 0.73, despite being locally stable at these points. Similarly, at y = --0.5, the growth rate is

179

,A 10"

I0.1

~0

~d'

id 2

0

~, -3 5)

/ ~:O.li / 2 -

Fig. 18. Resul ts of a t i m e - d e p e n d e n t ca lcu la t ion wi th prof i le as in Fig. 5, )~ = 8, k 0 = 0 .33, ~o ffi 0.1, 5 = 0.1. B o u n d a r y c o n d i t i o n s and init ial c o n d i t i o n s are the same as in Fig. 17. The local, y - i n d e p e n d e n t p r o b l e m is s tab le e x c e p t near y ffi - -0 .5 , where the m a x i m u m growth ~ate is 0 .11 ( m a r k e d by a d o t t e d line). However , t he fastest growing n o r m a l m o d e so lu t ion has a g rowth ra te of 0.73, whose g rad ien t is i nd ica t ed by the o t h e r d o t t e d line, peaked a b o u t y = - -0 .7 .

always much greater than the local value of 0.11. In this case, clearly, local calculations have no relevance whatever to the actual behaviour.

In summary, then, local solutions may, under some circumstances, describe the behaviour of certain initial value problems for a limited length of time. However, there seems no way of knowing what these circumstances are, for a given problem, wi thout solving the full eigenvalue problem; it is usually quicker to perform the actual spin-up calculation. What can be said with cer- tainty is that the long-term behaviour is that of the fastest normal mode at all the values of y, and that this will certainly dominate local effects where the local solution is stable. However, in the 1--2 e-folding times necessary for the solution to " lock on" to the normal mode behaviour, nonlinear effects are likely to become important where the normal mode has an appreciable ampli- tude so that even normal modes may have little relevance in actual problems.

This suggests that it may be necessary to rethink how to interpret and describe instabilities in many conditions: t ime-dependent behaviour-(almost certainly including nonlinearity) of an actual per turbat ion may be more relevant in some cases than the normal mode calculations discussed here; and both are far more relevant than local y- independent calculations in gen- eral.

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12. CONCLUSIONS

This paper does not set out to describe perturbations to a great many differ. ent mean-flow situations. Neither is it a guide to methods of solution of the quasigeostrophic equations. Because of this, not many detailed solutions appear here (and those which do, contain no discussion of such important things as eddy transport of momentum and buoyancy) , although many hours of computat ion were spent in efforts to ensure that no special cases had been missed.

Rather, this is an a t tempt to derive the kinds of instability, and the mecha- nisms involved, which should be expected in two-dimensional linear perturba- tion theory, as functions of k, ~ and ~. Exact details, for any given problem, require costly numerical solutions, but it is hoped that all the relevant instabil- ities observed for a given profile have been derived (with caveats in the transi- tion regions). Specifically, it has been shown that only ), and 5 exert qualita- tive control over the solution, k determines the length scale of the perturba- tions: this scales with the smaller of L and a, except for jet profiles in the baroclinic rdgime. On the other hand, ~ determines the degree of connect ion between the two layers (or, in the cont inuous case, between the surface and the deep water). Numerical confirmation of much of this work can be found in Holland and Haidvogel (1980).

Many very physical features have been neglected in this work; the most ob- vious of these will now be discussed. Firstly, mean profiles are not in general parallel and directed east--west. However, the mean-flow orientation is basic- ally irrelevant, since only the projection of the wave vector parallel to the flow is of importance * (Robinson and McWilliams, 1974). As a result, only the definition of ~ need be changed (to the gradient of f normal to the mean flow). This can give at most a quantitative change to the solution. If the mean flow is not parallel, however, then all depends on whether the mean flow changes appreciably over several eddy length scales. In the most typical oce- anic case, of large k, curvature of the mean flow over several hundred kilo- metres has little effect on eddies scaled on 30 km (k -1 ) but rather more on those scaled on 100 km ()-1/2) . In the atmosphere, k is less likely to be large enough for curvature effects to be neglected.

However, provided that )~ is large (or more precisely, that the eddy size is small compared with the length scale of variation of the mean flow) then one can adopt the two-scale formalism of McWilliams (1976). The solution devel- oped in Section 8, with the x axis parallel to the mean flow, is the valid first- order term in the actual solution (the local solution, as developed by Robin- son and McWilliams (1974) is not the first-order term in the actual solution). The higher terms show that the solution varies on the k -1 {phase) and k -x/2 (energy) length scales in both horizontal directions, i.e. a wave packet, con- fined spatially.

* Except that nonzonal flows do not possess a critical shear.

181

Secondly, mean profiles are seldom steady. The stability of Rossby waves has attracted much attention recently, beginning with studies by Lorenz (1972) and Gill (1974). By suitable coordinate transformations, the equa- tions of Gill (1974), or the baroclinic version, can be reduced to those for a perturbation to a steady parallel flow of the same form as used here, but with some third-order spatial derivatives. It seems likely that the techniques used here can be used to study the stability of such systems, although the basic effect of ~ and k will probably remain unchanged (cf. Kim, 1978). Other forms of unsteadiness are more problematic.

Thirdly, mean flows are seldom unbounded, at least in the ocean. It has long been known that boundaries stabilize mean flows. For large k, a vertical wall, or walls, far from the coalescence has little effect on the solution, the value of c being shifted slightly so that the integral (8.32) has a small real part, to bring ~ to zero at the wall. A wall in the vicinity of the coalescence (i.e. within O(), -I) of it) has more effect, but this is still qualitative. The most violent modification would be to the jet solutions, with a wall at the centre of the jet. This would remove the first (even) mode, and force the second (odd) mode into prominence. For large k the change in c is stabilizing, but still O(k - 1 ) and so is small. Therefore, boundaries normally have little effect if k is large; for smaller )~ the effects are stronger and can stabilize the flow (in the laboratory, for example). The vicinity of the Gulf Stream would be a typical area where wall effects would be important.

Fourthly, mean flows seldom have flat bo t toms beneath them. The modifi- cations to allow for a mean bo t tom slope are trivial. The y- independent growth rate in the two-layer case is strongly affected by bo t tom slope (Robinson and McWilliams, 1974) but not in the continuous case (Gill et al., 1974). How- ever, the effect is quantitative only, and may be retained in the analysis with- out any difficulty. Hence bo t tom slopes have no immediate effect on the conclusions.

One can speculate as to how these results apply to real geophysical systems, assuming that linear theory can give a reasonable guide to nonlinear phenom- ena (at least in certain areas, as noted in Section 11). Where, then, should one expect to find eddies produced by instability mechanisms in the real ocean or atmosphere? For reasonably large )~ (2 or 3 will usually suffice), one typically needs to find points where dc/dy vanishes ',coalescences). The simplest such point will be at jets, such as the Gulf Stream, Kuroshio and other oceanic boundary currents, or the mid-latitude atmospheric jet. Of course, this is hardly a new suggestion; such areas are known to generate eddies. However, these jets can generate eddies not because of their strength, or rapid growth rates, but because the axis of the jet is automatically a coalescence point *

* There are probably good reasons why geophysical jets have k ~ 1 to 2. When ~, ~> I, the eddy momentum fluxes can be derived exactly, and tend to narrow the jet, reducing k. Similarly, if k ~ 1, these fluxes tend to widen the jet.

182

Non-jet coalescences would be more difficult to find, and rather few in number, as suggested by the examples in Section 8. The shear would be much weaker than in jet regions, resulting in lower growth rates and weaker eddy activity.

Support for this exists both theoretically and observationally. Holland and Lin (1975) analysed the results of a two-layer fine-resolution grid point model of the spin-up of an ocean. The potential energy transfer from mean to fluc- tuations (the relevant quant i ty since the instabilities are baroclinic) was com- pletely centred on the model Gulf Stream and decayed rapidly from it. There appeared to be no other sources of available potential energy for the fluctua- tions, despite the fact that the mean flow was (locally) unstable over most of the ocean. The same conclusions were found in the multi-level model of Semtner and Mintz (1977).

Similarly, Dantzler's (1977) analysis of eddy potential energy levels in his- torical North Atlantic data shows only a very few isolated maxima of poten- tial energy outside the immediate Gulf Stream. He notes that they are in the vicinity of topographic features which may induce eddying (cf. Owens and Bretherton, 1978). It may also be that the mean flow is modified by the topography to produce a coalescence nearby. In any case, eddy production in the Atlantic does not appear to be a continuously occurring spatial phe- nomenon, but is confined to certain regions as speculated here.

ACKNOWLEDGEMENTS

A debt of gratitude is owed to Julian Smith, who computed in parallel with the author on many of the calculations leading to the conclusions in Section 8. Without his ability to program almost in real time, this work would have taken much longer. Thanks are also due to Adrian Gill for many critical discussions, and to Michael McIntyre, for impetus and discussions leading to the work in Appendix A. The work was supported by a grant from the U.K. Natural Environment Research Council. The referees have substantially im- proved the presentation of this paper, in addition to finding the solution (7.8); my thanks to them.

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