Jun 19, 1997 - specific case of linear shear flow over a top-hat ridge profile. In all cases modes are unstable over significant ranges of the relevant parameters.
Journal of Marine Research, 5.5,915-933,
Instability
in stratified
1997
rotating
shear flow along ridges
by G. A. Schmidt’ and E. R. Johnson* ABSTRACT This paper examines the interaction between continuously stratified horizontal shear flow and topography. It investigates when topography whose slope changes sign (i.e., a ridge or a trench) destabilizes a shear flow. The instabilities have the form of topographic Rossby waves and rely on the background potential vorticity gradient to exist. Barotropic flow is examined in detail for general horizontal shear over stepped topography and the longwave stratified case is examined in the more specific case of linear shear flow over a top-hat ridge profile. In all cases modes are unstable over significant ranges of the relevant parameters. Stationary unstable modes are seen to correspond to points where two opposing waves are brought simultaneously to rest by the shear flow and propagating unstable modes are seen where opposing waves are brought relatively to rest. Another interpretation of the instabilities in terms of resonance between waves of oppositely signed energy is also explored. Although the greatest growth rates for instabilities occur for the fundamental mode (or external Kelvin wave) interactions, for realistic oceanographic parameters the interactions of the slower moving, higher modes are more likely to be important.
1. Introduction Since shearflows of the form u = (uO(y, z), 0,O) are often exact solutions of the Euler equations,their stability to linear and nonlinear perturbations has been the subject of a great deal of study. While such shear flows in barotropic (or continuously stratified) rotating fluids have been studied extensively, relatively little attention hasbeen paid to the effects of topography. This is of major importance if the instability of shearflows is to be discussedin an oceanographiccontext. Generally, the highest shearsfound in the oceans are near either continental shelvesor other sharp topographic features such as submerged ridges or seamounts.Figure 1 gives cross-sectional profiles, from the Fine Resolution Antarctic Model (Webb et al., 1991) of the velocity field near the South Pacific mid-ocean ridge. Strong shearabove the ridge is particularly evident. The presenceof topography in a rotating flow is dynamically important becauseit allows extra, Rossby,wave modesto exist. In the absenceof shearthese modespropagate along any topography that is sufficiently “long” compared to their wavelength. Topographic wavespropagatebecauseof the variation in background potential vorticity which depends on rotation, topography and the shear.They travel with higher potential vorticity to their 1. NASA/Goddard Institute for Space Studies, 2880 Broadway, New York, New York, 10025, U.S.A. 2. Department of Mathematics, University College London, Gower Street, London, England, WClE 6BT. 915
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[55,5
Ai?oe1o.w 2.00~1000
_/ 0.50-2.00 i.,..j ::z: :: i_l .lO.oo. -2.M
A.."
03’s) 30
40
50
60
65
70
AtQw
. ..:
:;
i
j
to.00
zoos
r0.w
0.50.
2.w
Qso* 9.00~
i_i
40.00~
m
&wow
0.50 a50 4mo -lo.w
Nmm
Figure 1. Cross-sectionalprofiles from the output of the Fine Resolution Antarctic Model showing the cross-trackvelocity field in the vicinity of the South Pacific mid-ocean ridge. The sectionsare at 150E and 135E respectively (WOCE sections P16 and P17A). Note the almost-barotropic structureof the eddies and the large amount of shearpresent over the ridge.
right, so, for example, waves in a basin are always cyclonic, but can exist on both sides of a ridge, travelling in opposite directions. In the absence of shear it is easily shown that these modes are always stable. The addition of the shear has two main effects: it changes the background potential vorticity upon which these waves depend and it advects the waves. This means that some waves may reverse their direction of propagation and, more
importantly for the question of stability, that two waves on opposite sides of a ridge may be brought relatively to rest. When examining the stability of three-layer stratified flow, Taylor (193 1) pointed out that this can be one of the main preconditions for instabilities to occur. This resonance mechanism is quite general for conservative systems where wave modes
arise at distinct discontinuities in the flow (whether due to stratification as in Taylor (193 1) or at discontinuities in depth here). The growth of unstable modes in a conservative system
relies on the existence of wave modes with oppositely-signed energies (Cairns, 1979;
19971
Schmidt & Johnson: Effect of topography on shear$ow
917
Ostrovskii et al., 1986). Resonant interactions between modes can be characterized by their energies: only resonances between two modes with oppositely-signed energy are unstable, since only this type of growing disturbance conserves the total energy. The modes need to have similar (Doppler-shifted) frequencies to resonate but do not have to be of the same form (see for instance Sakai (1989) where Rossby and Kelvin wave mode interactions are discussed). It can be shown that the energies of topographic wave modes are related to the topographic gradient, and hence that waves on each side of a ridge have oppositely-signed energies. Previous literature has considered separately the effects in a stratified fluid of shear and topography. Moreover, the particular effects when the topographic slope changes sign have not been remarked on previously. Collings and Grimshaw (1980a, b), Collings (1986) and Bidlot and Stern (1994) deal with the effect of piecewise linear shear on barotropic shelf waves and the instability of these waves in the presence of more general horizontal shear how. They show that topography can destabilize an otherwise stable shear how along a coastal shelf, though they consider in the main the modification of already unstable modes due to topography. Barotropic and baroclinic instability of shear flow are considered by Killworth (1980) in rotating, stratified flow but only in the absence of topographic effects. This paper considers the stability of horizontal shear flow, u = (uO(y), 0, 0) over an infinite ridge and provides analytical results concerning the stability of the disturbances in two distinct limits. The equations, after a suitable nondimensionalization, depend on a stratification parameter B = N,,DljZ and a Rossby number, A = Ulfz. Here IV,, is the buoyancy frequency, D, L are typical length scales,fis the Coriolis parameter and U is the typical shear flow velocity. The Rossby number measures the importance of the horizontal shear relative to the rotation. The first limit considered here is the barotropic limit (B -=K 1). A second case where semi-analytical results can be obtained is the continuously stratified longwave, low-frequency limit. In the absence of topography the stability of this flow depends on the Rayleigh criterion (cf. Drazin and Reid, 1981) i.e. whether us changes sign. A more general condition including topographic effects is derived in Section 2 for the barotropic case along with an extension to the Fjortoft condition (Fjortoft, 1950), that z&u,, - c) must be somewhere negative. The analysis for a barotropic ocean becomes particularly simple (in terms of the barotropic stream function) when the ocean surface is taken to be rigid. This formulation is given in Section 2 and a number of results, which in the general case are rather complicated, are proved very straightforwardly. The rest of the paper considers linearly-sheared flow. The instabilities found are caused solely by the topography as uz is identically zero and so the flow does not satisfy the necessary Rayleigh condition for instability in the absence of topography. The general equations are derived in Section 3 (the barotropic case formally follows as the first term in an asymptotic expansion for B -=x 1). The barotropic equations are solved (Section 4) for the two cases of a top-hat ridge and a multi-stepped ridge. For the top-hat ridge a simple
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algebraic expression gives the phase speed which is either purely real or purely imaginary (corresponding to a stationary instability). The region of parameter space with unstable waves straddles the lines where dispersion curves, for waves concentrated at the two depth discontinuities, would cross in the absence of coupling. This region is also well delineated by the set of points where the waves that would exist without the shear but with an equivalent background potential vorticity would be brought to rest by the shear flow. The multi-stepped profile allows other modes to exist (one for every discontinuity of height) and instabilities are shown to exist for interactions between all modes. A mode that interacts with an opposing mode of the same number gives rise to a stationary instability (assuming the ridge profile is symmetric), while differing mode numbers give propagating instabilities. Section 4 also discusses continuously-stratified flow along a top-hat ridge in the longwave limit. The number of modes is infinite but the external Kelvin wave (EKW) is the most important, corresponding for small B to the fundamental mode of the barotropic case. In the rigid lid limit, this mode travels infinitely fast. The results indicate that the instability in all cases occurs when equivalent waves (those that would exist without the shear but with an equivalent background potential vorticity) are brought relatively to rest with an opposing wave on the opposite side of the ridge. Growth rates are largest when one of the modes is the fundamental mode and the greatest rate occurs when the two fundamental modes are brought absolutely to rest. In parameter ranges that are typical of ocean conditions, though, the fundamental mode (both in stratified and barotropic flows) moves much faster than typical shear velocities. This implies that the important wave-wave interactions will be among the higher modes. As an example in Section 5, the parameters are chosen to roughly represent the flows seen along the South Pacific mid-ocean ridge (6OS, 150W) of Figure 1.
2. Barotropic
formulation
The essential principle used in this formulation is the conservation of potential vorticity written in terms of the barotropic stream function. The governing equations are (Pedlosky, 1979):
(2.1) V.(h*u*)
= 0,
where u* = (CL*, v*) is the horizontal velocity vector, DIDt = dldt + u*.V, E* = vz - u; is the vertical component of relative vorticity, z * = - h*( y*) is the depth profile andfis the Coriolis parameter. A rigid lid has been assumed. Consider the perturbation field to the shear flow Uu&y*), where U is a typical velocity scale. Scale X, y with a typical length scale L (see the next section for the complete details), define 9, the stream function in
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Schmidt & Johnson: Effect of topography on shearpow
terms of the nondimensional depth h,
horizontal
perturbation
‘Py = -hu,
velocity components
‘Px = hv.
919 u, v and the
(2.2)
Writing the vorticity and velocity in terms of 9 and linearizing about the basic flow gives (2.3) where A = U&L is the Rossby number. Consider a wave-like disturbance travelling and frequency w. Eq. (2.3) then becomes
along the topography with wavenumber
k
(2.4) where c = o/k is the wave speed and Yr(x, y, t) = e(y) exp (iot - ikx). Standard manipulations (see for example Pedlosky (1979) or Ripa (1989; 1991) for a more generalized Hamiltonian treatment) yield the modified Rayleigh condition that ((Xub - 1)/h)’ must change sign for instability and the modified Fjortoft condition that Xu,((Xub - 1)/h)’ must be somewhere negative for instability. These conditions also apply to coastal shelves and imply that, if the shelf profile is monotonic, linear shear, u. = ?y, is always stable. For a ridge (where h’ < 0 for y < 0 and h’ > 0 when y > 0), the Fjortoft condition implies that positive linear shear u. = y is stable so long as h(X - 1) > 0 i.e. for small to moderate shear. For a trench, the conditions imply that negative shear is stable as long as X(X + 1) > 0. So in the northern hemisphere (f> 0), we should expect instability over a ridge to occur for negative shear flows. The topographic waves that could be supported in the absence of any shear how travel with the crest of the ridge to their right (i.e. for waves concentrated in y > 0, c > 0). Hence, the above conditions imply that moderate shear flow can only be unstable if the shear flow opposes the direction of propagation of the topographic waves. For larger values of shear (/Xl > l), the background potential vorticity gradient is reversed, and so the topographic waves will travel in the opposite direction and the instability could occur for the oppositely signed shear flow.
3. Equations of motion: Stratified
case
Consider an infinite ocean of depth D with a submerged ridge of width of order 2L (Fig. 2). Take Cartesian axes Ox* along the ridge, Oy* out to sea and Oz* vertically and let the ridge profile depend on y alone. Assume a uniform Coriolis frequency f and let the flow
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Figure 2. The domain A, with the z coordinate vertically upward, y perpendicular along the ridge. The width of the ridge is 2L and the open ocean depth D.
to the ridge, and x
be Boussinesq and incompressible with total density pg(z*) + p*(x*, t) and pressure &(z*) + p*(x*, t). Introduce the buoyancy frequency Nti2(z*)
= - ; $,
(3.1)
with the constant N,, chosen so the maximum value of N(z*) is unity. With no loss of generality, we take the shearflow as negative so that it opposesthe direction of the ridge waves on both sidesof the ridge. As in Section 2, considersmall wave-like perturbationsof frequency ofto the linear shearflow - Uy*IL and introduce the scalings (x”, Y”, z*, t*> = 6% LY, Dz, t!f), (u”, v*, w”) = (- Uy + EU, EV, (~cof*L/N; P” = --WLlGY2
+ @P$P,
D)w),
(3.2)
P* = (EP*&W>P~
where Eis the (small) magnitude of the disturbancevelocity. A detailed discussionof these scalingscan be found in Johnson (1991). They follow by requiring that the pressureis dynamically important in the momentum equations and from retaining the most general low-frequency density equation. The nondimensionalequationsbecome Du
E
W’&)*
- (A + 1)v = -px,
(3.3a)
Dv E + u = -py,
(3.3b)
Dw ot = -pz
- ~3
(3.4)
19971
Schmidt
& Johnson:
Effect of topography
DP --
oN2w
Dt
= 0,
u, + vy + (o/B2)w,
where B = N@”
921
on shearjow
= 0,
(3.6)
measuresthe importance of stratification in the dynamics and DIDt
=
aiat - hyaiax. The flow is hydrostatic with 1 D(P,)
w=---
P = -Pz’
oN2
Dt
(3.7)
’
provided the incident frequency, OL is small compared to the buoyancy frequency, No. Cross-differentiating gives asthe governing equation
(3.8)
If we consider a disturbance with wavenumber k and look for solutions proportional to exp (id - ikx), Eq. (3.8) can then be written in terms ofp alone. ttc + hY)P, + P) = 0.
(3.9)
where c = w/k is the wave speedand r(y) = (A + 1) - k*(c + AY)~. The free surfaceboundary condition is a2p, + N2(0)B2p
= 0,
(3.10)
(2 = 0)
wherea = d-gD/jL is the external nondimensionalradius of deformation. The vanishing of the normal velocity at the lower boundary z = -h(y) implies (o/B’)w
= -A’,
(3.11)
tz = -h(y))
which becomesin termsof the pressurealone (c + Xy)rp,
= -h’B2N2((c
+ Ay)py + p),
k = -h(y)).
(3.12)
In the far fieldp - constant as(1~1- a). In the limit of weak stratification (B
IYI> 1, IYIs 1.
(4.1)
This implies that h’ = 0 almost everywhere and simplifies the equationsenormously. The shearflow is taken to be linear in all subsequentanalysis. The equation for the barotropic streamfunction (2.4) becomes IJJ” - k2Q = 0,
(4.2)
almost everywhere with the condition at infinity that 4 should be bounded. The general solution of (4.2) is a linear combination of the two exponential functions ekyand e-b. We assumea similar solution in each of the three regions (y < - 1, - 1 < y < 1, y > 1) and usemasscontinuity to give an expressionfor the pressuredifference between the regions. Thesetwo expressionscan be written (Cairns, 1979) cosech(2k)
[ply=+, = A+Q(w W + A-WC + A>
h
, (4.3)
cosech(2k) [P],=-~
= A-D2(m,
k) - A+k(c
- X)
h
,
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Schmidt & Johnson: Effect of topography on shearjow
923
where D,,, are the dispersion relationships (4.4) for waves concentrated at one discontinuity of depth, assuming a rigid boundary at the other discontinuity. The constants A+ and Arefer to the amplitude of the decaying stream function as y - 00 and y - --00 respectively and the second term in each expression is a measure of how closely coupled the waves are. The dispersion relations can be derived as D,,*(0,k)=(l+h)~~-I)i(0ihk)jCOthho
+I).
(4.4)
Requiring the pressure difference to vanish at y = ? 1 gives the dispersion relation for the full system as D(o, k) = D,(o, k)D,(w,
k) + k2(c2 - X2)
cosech2 (2k) h2
= ”
In terms of the phase speed this is c2 _ A2e-4k(hk + 1 + A)2 - (Xk - A(1 + X))2 9 k2(A2e-4k - 1)
(4.6)
where A = (1 - h,)l( 1 + h,). Only the two fundamental modes (one travelling in each direction) are present since the top-hat profile filters out all others in the barotropic limit. The wave speed for the fundamental mode is either purely real or purely imaginary with the two regions separated by a neutral curve in (k, A, X) space (Fig. 3). Since instabilities correspond to purely imaginary phase speeds, the corresponding disturbances are stationary. The phase speed is singular at k = 0 since, in the longwave rigid lid limit, the fundamental mode travels infinitely fast. As expected from Section 2, instability is confined to the region A > 0, i.e. to ridges. The instabilities present for relatively strong shear (A = 1) are shown in Figure 3a for varying ridge height (trench depth). Figure 3b shows that as A increases from 0 (i.e. as the shear flow becomes stronger), longer waves become unstable. At any positive value of A, there are modes that are unstable, however for very short waves the growth rates are small. For a given shear and height modes are unstable over a range of wavenumbers. For increasing height or shear the unstable waves become shorter and the range of wavenumbers narrows. There are also unstable modes for A < - 1. These correspond to very large positive shear where the background potential vorticity gradient has changed sign. These large values of shear dominate the background rotation and are unlikely to be seen in the oceans. The energy needed to excite each mode (with a rigid wall replacing the other discontinuity) is proportional to WC~D~,~/&II (Cairns, 1979). Since D1 and D2 are related through the transformation o :- -w, these energies are oppositely signed. Here (4.7)
924
Journal of Marine Research
.l
4 0.0
0.5
1.0
1.5
wavenumber (k)
2.0
00
0.5
1.0
2.0
wavenumber (Ii;
Figure 3. The results for the phase speed in the barotropic top-hat casewith linear shear.(a) The variation of c2with A = (1 - h)l( 1 + h), where h is ratio of the height of the ridge (or depth of the trench) to the open ocean depth, and k, the wavenumber,for A = 1. (b) The variation of 8 againstA and k for fixed h = 0.5. Regions of instability are shaded.As expected there is no instability for A < 0, correspondingto a trench. The dotted line showsthe points where the equivalent wavesthat would exist without the shearflow would be relatively brought to rest by the shear (at y = t 1). The dashed line shows at which point there would be a resonancebetween two waves with oppositely-signedenergies.
If the coupling is small in (4.5), the solutions to D(o, k) = 0 are close to those of &(o, k) = 0. When the solutions of the two separate dispersion relationships are close, there will be complex (i.e. unstable) solutions of the full relationship only when the energies defined above are of opposite signs. When they are of the same sign, as would be the case for waves on a two-stepped coastal shelf, the dispersion curves do not meet but instead “kiss” (cf. Schmidt and Johnson, 1993). The dotted lines in Figure 3 are the points at which the dispersion curves for the two waves considered separately would cross. Another way of examining this resonance feature is to consider the waves that would be present without the shear flow. Since the shear flow alters the background potential vorticity, “equivalent” waves are those that exist when the background PV is (A + 1)/h. The dashed line in the figures represents the points where the phase speed of the equivalent waves exactly matches that of the shear flow and hence the two opposite waves (concentrated at y = t 1) would be brought relatively to rest at the discontinuities. This is not exactly analogous to considering the waves individually as above, though each procedure gives very similar results. Both lines lie close to the line of largest growth rate (for fixed k) and support the resonance mechanism for instability. ii. M.&i-step profile. Johnson (1990; 1993) notes that long waves over complex ridge/ trench profiles can be approximated arbitrarily well by replacing continuous topog-
19971
Schmidt & Johnson: Effect of topography on shear$ow
925
raphy by stepped profiles. For the present case of a ridge this corresponds to considering the profile IYI > 1, h(y) = ;’
n,
Yn-I < Y -=I Ym
n= 1,2,...,N
whereh, is the depth between ynpl and yn and N is the total number of steps.As previously h’ is zero almosteverywhere so that in the barotropic caseat least, the solution above each step can be immediately written down. For this discrete problem the exact (numerical) answercan be found and increasing N gives an increasingly accurate description of the flow over arbitrary profiles. The solution follows as in Johnson (1990). The governing equation is the same as for the top-hat ridge profile (4.2). This has the solution $, = a, cash . ky + b, sinh ky over each stepand a0exp (-ky), b. exp (ky) for y > 1 andy < - 1 respectively. The jump conditions at y = -’ 1 can be used to eliminate a0 and b. and the continuity of Q at y = yn can be usedto progressively eliminate b,, IZ= 1, . . . , N - 1. This leavesN + 1 variables and N + 1 equations which dependon c linearly: a system which can be solved using standardtechniques.There is no restriction on the spacingof the y,, so more points can be addedwherever there is a needfor greater definition. For the specific example of the sinusoidalprofile h(y) = 1 - h,,, sin (rr(y + 1)/2) and evenly spacedy,, the first few wave speedsconverge rapidly asN increasesand it suffices to take N = 20 in the following examples. In Figure 4, X = 0.2 and k = 1 while h,, varies over [ - 1, 11.Since the ridge is symmetric about y = 0, waves occur in pairs. For any wave its reflection about y = 0 is also an allowable wave travelling with the samespeedin the opposite direction. Plotting Re(c2) collapsesboth stable pairs and stationary instabilities onto the sameline simplifying the figures. Stationary instabilities occur when a wave pair is resonant and are shown by negative values of Re(c2). Propagating instabilities occur when the resonanceis between different modes.There are no instabilities for negative h,, (i.e. trenches)Over ridges (h,, > 0), there are both stationary and propagating instabilities. The growth rates for stationary instabilities are largest, having here doubling times of the order of days. It is not as straightforward to predict these results as in the previous example using a top-hat ridge. The equivalent waves can no longer be considered to be localized at a discontinuity of height and soit is difficult to assigna single speedmeasuringthe effect on the wave of advection by the background flow. Nor is it simple to generatethe analogueof (4.3-5). However, the broad featuresof the resultscan be explained using the samegeneral argument.The fundamentalmode is stablefor sufficiently long waves but asthe wavenumber increasesthe speedof the mode decreases.It then interacts with the higher modeson the opposite side of the ridge and destabilizes each in turn. Note that if the surface were taken to be free, the sole qualitative difference would be the that the fundamental mode speedwould remain finite in the longwave limit. This would have little bearing on the structure of the problem at finite k.
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Wave speed (Re(c*))
rate (Im(c)) ““7
hmax
I\
h ma*
Figure 4. (a) The real part of the wave speed squared and (b) the imaginary part of the wave speeds for the 20-step approximation to a sine shaped ridge profile in the barotropic limit plotted against the height of the ridge h,,. The wave number is k = 1 and the Rossby number A = 0.2. As expected there are no instabilities for h,, < 0. The fundamental mode is stable to longwave disturbances and dominates the stability diagram. Instability is apparent when c2 is negative (a stationary mode) in (a) or when c is complex (a propagating mode) as shown in (b). The largest growth rate is for the fundamental mode when h = 0.2.
b. StratifiedJlow
in the longwave limit
For simplicity, let the stratification be uniform, N(z) = 1, and take X + 1 > 0. Resealing the horizontal coordinate as Y = ly, where 1 = JmlB, in (3.13) gives PYY
(4.9)
+ Pzz = 09
with the resealed boundary condition w + WP, where K = lc and H(Y) become
= -ff’(YMK
+ WPY + P>,
(z = -ff(Y)h
(4.10)
= h(Zy). For the top-hat profile (4.1) the boundary conditions
a*p, + B*p = 0,
(z = 01,
pz = 0,
(z = -h,,
p + (K + XY)p, = 0,
-l>,
(4.11)
(Y=tl,-l 1) or hyperbolic (cash and sinh terms for 1Yl < 2). Hence the general solution in each region can be written as an infinite series
m c a, exp(-MY - OR(z), n=O I
Y>l
cash (Y Y
sinh a,Y + 4 sinh p=(iin=O ‘7,coshan/ n n
n(z),
I YI -=c1,
(4.13)
m
b, exp(MY + OP,kh Ix \n=o
Y< -1,
where a,,, b,, c,, d,, are constants to be determined from the matching and boundary conditions at Y = ? 1.These imply that the mass flux, p + (K + XY)p, = 0, and pressure, p, are continuous for z, E [-ho, 0] at Y = 21. The functions A,(z) and the functions Cn(z) = cos (nn(z + l)/( 1 - ho)) are complete and orthogonal over z E [-ho, 0] and z E [ - 1, -ho], respectively. Multiplying the matching equations by these (normalized) functions and integrating over their ranges in z gives a set of simultaneous equations in the constants a,, b,, c,,, d, and the wave speed K. After eliminating cnr d, these equations become 5 Zm,(a, + b,)(ol, tanh a,2 + p,) = 0, n c
n
Lb,
- b,)(~,
+ P, tanh cw,Z) = 0, (4.14)
for m = 0, 1,2 . . . , where I,, = .&, A,(z)B,(z) dz, and J,, = j’:y C,(z)B,(z). Truncating the expansions leaves a standard linear eigenvalue problem. As in the barotropic case, instability can be predicted by using either the dispersion curves of each set of waves at a discontinuity separately (complicated due to the breaking of symmetry), or by calculating
928
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of Marine
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the phasespeedsof waves that would exist without the shearflow but with an equivalent background potential vorticity (writing Bequiv = BIdm and solving the “no shear” equations). In parameter ranges that are typical of oceanic conditions, the external Kelvin wave (EKW) mode travels considerably faster than the internal modesand any reasonableshear flow. It is convenient to take first parameter values corresponding to reasonably large shears.These demonstratemore clearly how the EKW interacts with the higher modes. Later examples show the behavior for values more relevant to the Southern Ocean. Figure 5 gives resultsfor ho = 0.5, a = 4 and X = 1 and the stratification parameterB in the range [0, 51.As in previous figures, both the real part of the wave speedsquared(Fig. 5a), and the imaginary part of the wave speed(the growth rate of the unstablemode) (Fig. 5b) are plotted. Once again, the strongestinstabilities in the flow are the EKW interactions which, in the barotropic limit B - 0, match the fundamental modesin the barotropic (free surface) case in the limit k - 00.In the barotropic limit the internal modesare concentrated in a boundary layer of width O(B) at the discontinuities and are advected with the shear flow (Re (~2)- X2). As the stratification increases, the stationary instability of the EKW disappearsand a seriesof propagating instabilities appearsas the EKW interacts in turn with each of the higher modes concentrated on the opposite side of the ridge. For even stronger stratification the first internal mode will interact with oppositely-propagating waves to produce first a stationary instability and then another series of propagating instabilities (not shown). If the fluid surface had been taken to be rigid, the fundamental mode speed would be infinite and the figures would show singularities where the fundamentalmodechangedstability. It is therefore valuable to retain free surfaceeffects in the longwave limit, even when the external Rossby radius is large, to regularize the problem. The growth rates for the stationary unstable modes decrease sharply with increasingmode number and the growth rates for the propagating modesdecreaseas both the interacting modenumbersincrease. Insight into this pattern can again be gained by considering the equivalent background potential vorticity waves. Only the EKW mode travels with a nonzero phasespeedin the barotropic limit and as the stratification increasesthe phase speedof of any given wave increases.In the example given here, the two fundamental modes are simultaneously brought to rest by the shear flow at B = 1.0 and this is close to the value for the fastest-growing stationary mode.With increasingB the speedof each wave pair will match, in turn, the speedof the shearflow at the discontinuities. Stationary unstablemodesoccur at each of these points. If the stratification is increased further after such a stationary instability, a seriesof propagating instabilities forms. At these points there is at least one pair of modesthat are travelling faster than the shearflow and propagating in the opposite senseto all higher modes.As the stratification increases,the speedof these modes will match, in turn, the speedsof each higher mode on the opposite sideof the ridge. The shear flow brings two opposing modesrelatively to rest. By examining the equivalent waves, this is predicted to happenat B = 3.6 between the fundamental mode and mode 1, at B = 4.0
Schmidt & Johnson: ESfect of topography on shearJlow
Stratification
i 1 I 1.25
(B)
929
II 1 375
2.5 Stratification
I 50
(6)
Figure 5. The variation with stratification of (a) the real part of the square (b) the imaginary part of c, the wave speed, in the longwave approximation. Here the Rossby radius a = 4, the Rossby number h = 1 and the height of the step ho = 0.5. The external Kelvin wave mode (EKW) is unstable as (B - 0) and matches the barotropic fundamental mode. All the other modes are advected with the shear flow in that limit. As the stratification increases the speeds of all the higher modes increases and hence match the speed of the EKW mode and become unstable in turn. In (b), the dotted lines are where the instabilities are predicted to be assuming that two equivalent waves are brought relatively to rest. between the fundamental mode and mode 2 and at B = 4.2 between the fundamental mode and mode 3. As can be seen in the figure these points provide very good estimates for the centres of the propagating instabilities. This pattern of a stationary instability of a mode pair followed by a series of propagating instabilities between different modes also occurs in the previously discussed example of a multi-stepped ridge in barotropic flow. A similar resonance mechanism thus describes the results there even though no resonance condition can be easily calculated. 5. Application
to the South Pacific mid-ocean
ridge
The mid-ocean ridge in the South Pacific is in the main path of the Antarctic Circumpolar Current between 5% and 65s. It runs roughly east-to-west between 135E and 150E. It rises around 2000 m above the sea floor at a depth of 40004500 m and is between 800 km and 1500 km wide. Estimates of the current shear across the ridge were obtained from modeling results from the FRAM (Fine Resolution Antarctic Model) group (Webb et al., 1991a, b). This eddy-resolving general circulation model has been used to study the southern oceans and results show a great deal of eddy activity concentrated above the ridge over the whole water column (Fig. 1). These features are coherent for around 300-500 km. (according to the modeling results) and hence have nondimensional wavenumbers of 0( 1). A recent analysis (Ivchenko et al., 1997) of the kinetic energy budget in FRAM has indicated that there is a substantial transfer of mean kinetic energy to eddy kinetic energy in the model domain and that this is assumed to be due to barotropic instabilities.
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As a realistic example of the solutions derived in Section 3 the dimensional quantities are taken asf = 1.2 X lop4 s-i, L = 400-700 km, D = 4-4.5 km, No = 4 X lo3 s-i, and h = 0.5. This makes the stratification parameter, B = 0.2-0.35, and the nondimensional Rossby radius of deformation, a = 2.5-4.5. The most important parameter, X, depends on the shear present. Assuming an absolute maximum of 1 m s-l of velocity difference across the ridge, A should lie in the range [O, 0.021. Since the shear is positive (assuming a coordinate system as in Fig. 2) and the ridge is in the southern hemisphere, this is equivalent to takingfas positive and the shear as negative as in the examples above. Figure 3b suggests that the fundamental mode will travel too fast to be destabilized by the relatively small shear in this particular example. Allowing for a smoother profile, by taking the multi-step profile of the previous section, introduces higher topographic modes that travel slower than the fundamental mode and can therefore be destabilized by weak shear. Figure 6 shows that for X = 0.01, unstable modes exist for multiple wave numbers although the growth rates are smaller than in the higher shear case depicted in Figure 4. Typical doubling times for disturbances are around 60 days (at k = S), or 100 days at k = 3. Since the stratification is not completely negligible, the stratified longwave solution may also be relevant. Taking B = 0.3, we can see in Figure 7 that for small X, it is the interactions of the higher modes that are important (the EKW mode travels much faster and is not shown on the graph). The bands over which each pair of modes are unstable are quite narrow. The longwave must be extrapolated to finite wavenumber to give an estimate of the growth rate of any instability. Taking k = 1 as a typical value gives a doubling times for disturbances of the order of one hundred days or more. These estimates lend support to the idea that topographic wave resonance may be responsible for the eddies seen in the FRAM modelling results. The growth rates calculated in the barotropic, multi-step topography case and the top-hat ridge, longwave stratified case are however small and a wider, and considerably more complex, investigation including the combined effects of finite wavenumber, complex ridge profiles and stratification is needed to determine whether resonances can affect the flow over shorter periods. The above results suggest that these effects reinforce one another rather than compete and so a full investigation could indeed find more rapidly growing modes. 6. Discussion Instabilities have been demonstrated in a horizontal shear flow, that would otherwise be stable, when the current interacts with bottom topography. As well as the topography profile, h(y), three non-dimensional parameters are important in determining the stability of the flow: the stratification B, the Rossby number X and the radius of deformation a. Generally, the stability will also depend on the the wavenumber k of the disturbance. The cases discussed here are (i) barotropic flow with a rigid lid (B + 0, a >> 1) over top-hat and multi-step profiles, and (ii) longwaves in continuously stratified flow (w, k < 1) over a top-hat profile.
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25
2.5
5.0
wavenumber
(k)”
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I 5.0
wavenumber
I 100 (k)‘.5
Figure 6. The results for the 20 step approximation to a sine-shaped ridge profile in the barotropic limit with ridge height h,,, = 0.5 and Rossby number A = 0.01. The instabilities are resonances between the higher modes, and have their largest growth rates at wavenumber k = 3.5 and k = 8. The time for such a disturbance to double is between 60 and 100 days. The shear (X) is allowed to vary arbitrarily. In the absenceof shear(A = 0) all waves are stable whatever the profile or (statically-stable) stratification. The top-hat profile in the barotropic limit filters out all modesexcept the fundamental mode. With more complicated topography, for instance an N-step profile, there are N + 1 (the number of height discontinuities)modeswhich are presentfor all wavenumbers.Generally as the wavenumber of the modesincreasesthe phasespeedwill decrease.For stratified flow, internal modes
A = URL
h = UAL
Figure 7. In the stratified longwave limit there are many interactions between higher modes, although the bands over which the modes are unstable are narrow. Growth rates seem to be too small to be relevant in the open ocean.
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are possible and as the stratification increases, the speed of all the modes increase. The external Kelvin wave mode in stratified flow matches the fundamental mode in the barotropic limit. However, in the longwave limit, the fundamental mode speed remains finite only with a free surface. The fundamental/EKW mode travels significantly faster than the other modes (given realistic parameters). A shear flow alters the background potential vorticity and advects the waves. For top-hat profiles waves can be approximated as being concentrated along the lines of depth discontinuity. If the velocity of the shear flow at these lines is much greater than the speed of the waves, then the waves will be swept along with the flow. It is only when the velocities of the shear flow and of the mode are comparable and opposite that the possibility of unstable modes arises. A convincing model for the onset of instability describes the resonance between two different waves that the shear flow has brought to a similar velocity. Two methods of determining when this might occur have been used in this paper. First, the waves that would exist at each discontinuity when the other discontinuity is replaced by a rigid wall can be examined. Where the independent dispersion curves cross, a resonance can be expected. Second, the speed of the waves that would be present in the absence of the shear flow, but with an equivalent background potential vorticity can be calculated. For instance, in the barotropic limit, the background potential vorticity when the shear is imposed is (U/L + f)lh so then the equivalent waves with no shear will have a new “rotation” parameter of U/L + $ In the stratified longwave limit the equivalent stratification parameter is given by B/d=. These equivalent modes can be calculated using the straightforward topographic wave equations. Resonance can be expected if the shear flow brings two of these equivalent waves relatively to rest. Neither of these two indications of resonance are sufficient to ensure instability. They will only do so if the conditions derived from (2.4) and (3.15) are satisfied, i.e. the background potential vorticity gradient changes sign and the shear flow is in the opposite sense to the direction of the wave modes. The energy of the modes is related to these conditions, and resonance leads to instability only if the two modes involved have oppositely-signed energies. In the context of realistic oceanographic flow it seems likely that the mechanism examined here could be of importance in some areas. In the example explored in Section 5, the shear flow is too weak for the barotropic or longwave stratified calculations to provide anything more than an indication that these instabilities are present. However, growth rates for finite wavenumbers in stratified flow may be larger than those calculated here and may provide more conclusive evidence. The South Pacific mid-ocean ridge is relatively wide; other ridges, particularly the Lomonsov Ridge in the Arctic Ocean, are narrower and have significant shear across them. More data and/or higher resolution modelling studies of these regions would be needed to compare with the theory presented here. Acknowledgments. We would like to thank two anonymous referees for inspiring us to add some of the analysis here. We would also like to thank Professor L. A. Mysak for financially supporting GAS during the final stages of preparation of this article, and Dr. Andrew C. Coward, from the OCCAM
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group at the Southampton Oceanography Centre, for preparing Figure 1. Funds for this research were awarded under the FRAM Special Topic Award from the Natural Environment Research Council. REFERENCES Bidlot, J.-R. and M. Stem. 1994. Maintenance of continental boundary-layer shear, through counter-gradient vorticity flux in a barotropic model. J. Fluid Mech., 271, 55-85. Cairns, R. A. 1979. The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech., 92, l-14. Collings, 1. L. 1986. Barotropic instability of long continental shelf waves in a two-layer ocean. J. Phys. Oceangr., 16,298-308. Collings, I. L. and R. Grimshaw. 1980a. The effect of current shear on topographic Rossby waves. J. Phys. Oceangr., 10, 363-371. 1980b. The effect of topography on the stability of a barotropic coastal current. Dyn. Atmos. Oceans, 5, 83-106. Drazin, P. G. and W. H. Reid. 1981. Hydrodynamic Stability, Cambridge University Press, Cambridge, England, 525 pp. Fjortoft, R. 1950. Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex. Geofys. Pub]., 17, l-52. Ivchenko, V. O., A. M. Treguier and S. E. Best. 1997. A kinetic energy budget and internal instabilities in the Fine Resolution Antarctic Model. J. Phys. Oceanogr., 27, 5-22. Johnson, E. R. 1990. The low-frequency scattering of Kelvin waves by stepped topography. J. Fluid Mech., 215, 2344. ~ 1991. The scattering at low frequencies of coastally trapped waves. J. Phys. Oceanogr., 21, 913-932. ~ 1993. Low-frequency scattering of Kelvin waves by continuous topography. J. Fluid Mech., 248, 173-201. Killworth, P. D. 1980. Barotropic and baroclinic instability in rotating stratified fluids. Dyn. Atmos. Oceans, 4, 143- 184. Ostrovskii, L. A., S. A. Rybak and L. S. Tsimring. 1986. Negative energy waves in hydrodynamics. Sov. Phys. Usp., 29, 1040-1052. Pedlosky, J. 1979. Geophysical Fluid Dynamics, Springer-Verlag, Berlin-Heidelberg-New York, 710 pp. Ripa, P. 1989. On the stability of ocean vortices, in Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence, J. C. J. Nihoul and B. M. Jamart, eds., Elsevier Oceanographic Series, Amsterdam, 167-179. ~ 1991. General stability conditions for a multi-layer model. J. Fluid Mech., 222, 119-137. Sakai, S. 1989. Rossby-Kelvin instability: a new type of ageostrophic instability caused by resonance between Rossby waves and gravity waves. J. Fluid Mech., 202, 149-176. Schmidt, G. A. and E. R. Johnson. 1993. Direct calculation of low-frequency coastally-trapped waves and their scattering. J. Atmos. Ocean. Tech., 10, 368-380. Taylor, G. I. 193 1. Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Sot., A132,499-523. Webb, D. J., P. D. Killworth, A. Coward and S. Thompson. 1991a. The FRAM Atlas of the Southern Ocean. Natural Environment Research Council, Swindon, U.K. Webb, D. J. and others (The FRAM Group). 199lb. An eddy-resolving model of the Southern Ocean. EOS, 72, 169-I 74.
Received: 17 October; 1994; revised: 19 June, 1997.
