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On open ocean baroclinic instability in the Arctic
Deep-Sea Research, 1976, Vol. 23, pp. 637 to 645. Pergamon Press. Printed in Great Britain.
On open ocean baroclinic instability in the Arctic J. E. HART* and P. D. K1LLWORTHt (Received from first author 12 March 1975; revision receivedfrom both authors July i 975) simple linear model is used to investigate the possibility of baroclinic instability in the Arctic. The basic state consists of a barotropic flow directed along depth contours, a baroclinic component uniform in horizontal directions which decreases exponentially with depth, and an exponentially decaying stratification. With no ~-effect there is always a shortwave cutoff for wavelengths shorter than the radius of deformation based on the surface N ~and total depth. With [3there appears to be no shortwave cutoff but the growth rate peaks near the same values of wavelength. The results suggest that the 10- to 20-km eddies observed in the Arctic are probably not generated by baroclinic instability in the open ocean but may arise from instability in the shallow regions. Abstract--A
1. I N T R O D U C T I O N
RECENT observations of NEWTON, AAGAARD a n d COACHMAN(1974) and HUNKINS(1974) have shown the presence of substantial eddy activity in the interior of the Canada Basin of the Arctic Ocean. The eddies have horizontal scales from 10 to 20 km and velocities on the order of 30 cm s-1. We investigate here the hypothesis that the eddies are the result of a baroclinic instability process in the open ocean, or in shallow water near continental margins. Open ocean baroclinic instability would appear a priori to be a likely candidate for the observed eddies. This is because the observed eddy sizes agree well either with the first-mode Rossby radius of deformation calculated from the observed stratification (25 km), or with a radius of deformation based on the depth over which the stratification is important, which is, as Hunkins shows, about 17 km. However, both the [3-effect and the bottom topography can be essential elements of the instability process. The theory developed in Sections 2 to 4 shows that the unstable wavelengths, for realistic stratification and mean currents, scale with the parameter N(0) D where
f
unstable waves in the open ocean whose length scales are 10 to 20 times larger than the radius of deformation. It is thus concluded (Section 5) that if baroclinic instability produces the observed eddies, the instability must occur in the shallower parts of the Arctic Ocean where large shears can extend to the bottom. 2. THE QUASI-GEOSTROPHIC EQUATION
Most of the analysis in this paper concerns the stability of an inviscid quasi-geostrophic flow on an f-plane. For small wavelengths, order 20 km, the neglect of the [~-effectis apparently permissible; however, for wavelengths of order 100 km or more, [~ should be included, and in Section 4 the stability problem will be solved numerically including planetary vorticity advection. The governing equation is the familiar potential vorticity equation (with [3 retained for clarity, although it will often be set to zero):
d [ v n ' , + O ~c3hb+ [3y] dt
O,
(2.1)
Oz c3z
with d d-t ~ O-t + J(q;'
)"
(2.2)
N(0) is the Brunt-Viiis/il~i frequency just below *Department of Meteorology, Room 54-1713, Massathe mixed layer, f the Coriolis parameter, and D the total depth. (In fact, the radius of deformation chusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A. is also proportional to this factor, but with a -[-Department of Applied Mathematics and Theoretical numerical factor of about one-tenth.) This yields Physics, Cambridge University, Cambridge, England. 637
638
J.E. HARTand P. D. KILLWORTH
Asterisked variables are dimensional; nonasterisked variables are dimensionless. The flow is geostrophic, so that
where q~is the dynamic part of the pressure, scaled by p o f L U , and p is the deviation from the background density 15(z), normalized by P0. Velocities have been scaled by U, the scale of the mean flow whose stability is to be tested, and horizontal scales (x* east, y* north) by L. Depths (z* ; z* =- 0 at the surface) are scaled by D and time by L/U. Finally, ~ == ~*L2/U. When ~ ~ 0 the only parameter appearing in the interior equation is
f2L2po -- gD2-iOP-~-Z* i
"
(2.3)
The boundary conditions are that d (e+~ + ha ) dt
0
(2.4)
fL
+o = F(hB) + "~(x,y,z). To simplify the analysis we assume that hB contours are locally unidirectional so that the bottom slope is uniform over a typical wave scale and •qb 0 is effectively constant in x and y. This type of approximation was derived by a formal perturbation expansion for the two-layer model by ROmNSON and MCWlLLIAMS (1974). For arbitrary orientation of the horizontally uniform baroclinic basic flow component to the bottom contours, the interior equation (1.1) is satisfied if we choose ~ --ye b~. Thus the baroclinic component of the mean current is oriented in the x direction. We now perturb about this basic state, writing ~ ' ---- ,~?(z) e i("x~ 8y--
at z ---- 0 and z =~ --1. In this equation h B --= 0 at z ----0 and hs = h*/DRo at z = --1. Here h* is the height of the bottom topography relative to z* ---- - - D , and R 0 is the Rossby number U R o ~- ~ .
where b is determined by depth and generally takes a value of about 6 to 8. The stratification then decays to 2 to 10% of its surface value in about 1000 m. The basic velocity distribution is made up of both barotropic and baroclinic parts. The barotropic part is constrained to be parallel to the depth contours, so the basic state may be written
(2.5)
The perturbation makes an angle A : tan-l~/~ with the baroclinic basic flow Uo -- e b~. It can easily be shown that the presence of a uniform barotropic current Uo only affects the real part o f . , the frequency of the perturbation, not its growth or decay. If we write
= ~Ui,cos 0 + ~Ubsin 0 + 7c, These equations have been used by PEDLOSKY (1964) and many others in similar studies of where 0 is the angle between Uo and the depth baroclinic instability. contours, all reference to the barotropic flow is removed. The linearization of (2.1) and (2.4) about 3. THE BASIC STATE AND PERTURBATION ~b0, using the presumed forms for ~b' and ~ yields EQUATIONS a two-point boundary value problem for the A good representation of the stable density determination of ~c, the perturbation growth rate. stratification of the interior Arctic is achieved by A uniform topographic slope is included for writing generality. It is represented in the parameter S ---- ~h*/DR o in which ~h* is the upslope change e0 e0 of depth over a length L, D is the average depth, --~ z* = 0 . e , and R o is the Rossby number U/fL. With some ~z* ~z* [ b~
On open ocean baroclinic instability in the Arctic straightforward manipulation and transformation to the logarithmic coordinate rl = d '~, the stability problem for [~ : 0 can be reduced to
dq~ (1 - - c ) ~ = q ~
--c~
d~p
in effect the radius of deformation based on surface stratification and D. The general solution of (3.1) is q~ = 111/211 (2Tr111~) + B
d29
+ Qq~ = 0
atrl = 1
(3.2)
atrl --- 0.
(3.3)
639
TII'~
K1 (2T~1/~),
(3.7)
where the coefficient of the first term is set equal to 1 as a normalization. Using (3.2) and (3.3) and the recursion relations for modified Bessel functions we obtain an eigenvalue equation for c, c~T~ [K0(27) + I0(2T)M] + cy [--yK2(2y) -- M712(22') -- QIo(2y)/2V]
The parameters entering this eigenvalue problem are a non-dimensional wavenumber scaled on the surface deformation radius,
+ Q/~(2y) = 0,
(3.8)
where M ~- In (2?) -- b/2. ~ / =
b N / ( 1" + X'~ n ' A ")
,
(3.4)
where
4. RESULTS At large enough Y so that Ko,e(2y) ~ Io,,(2Y) the discriminant can be written as
= . f~L2P0
gD l°o l I&z* I o
(3.5)
and S cos 0 Q---(1 b~
+ tan 0 tan A) -- 1. (3.6)
In writing (3.3) we have assumed that Uo(z) = 0 at z = -- 1. The error in doing this is e-b which means that we may be omitting the possibility of unstable modes whose growth rates would be of order e -b relative to the growth rates found below. Note that the key parameters y and Q are independent of the scaling length L as is the dimensional growth rate **. L enters as the horizontal scale of the variations of the basic state. For our local analysis to be valid we must have = 2~tL/X* be large. However, the critical parameter in the stability analysis is y which is essentially a measure of the wave scale k* in units of 2~tL ~,bV'~ '
T'[MI2(27) -- Qlo(2y)/272] 2. Thus, at large ? (2 or 3 may be large enough because of the factor 2 in the argument) the flow will be stable. Thus this exponentially stratified and sheared flow exhibits a short wave cutoff. For small y the discriminant becomes 74[(1 + Q ) Z (1 + Q ) ( l n 2 y - ~) 4~,~ + + ~Q- + 0(74)] ~ Thus as ? l+ 0 there will be a long wave cutoff at finite T unless Q = - 1 . Finally, note that since there is no instability at large y, the quantity M is negative in that range of T for which instability is possible.'f Using the asymptotics of K0 and I o it is possible to show that for our values of b, unstable modes can only be found for negative Q. Thus, there would appear to be a fairly small range in ~fWe assume b is _> 4.
640
J.E.
HART a n d P. D . KaLLWORWrt
-10 -
0
Q
-I.0
-.I
0
.5
I
L
J
1.0
k5
20
Fig. 1 (a) -I0 -
(~ -L0
2 ~
©
59 . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
1.0
] .................
15
2.0
Fig. 1 (b) Fig. 1.
C o n t o u r s o f c~ (a) a n d c, (b) for b
parameter space where instability is possible. Figure 1 shows isolines o f c i and c, as functions o f Q and y for b = 6. The general asymptotic results are reflected in the c I diagram. It is seen that the largest ci occur near 7 = 1/2 for Q = - 1 . For Q near - 1 the largest c~ occur for small V but recall that the actual growth rate is :{ci so that as
6, H = 0 (inviscid).
T oc :{ the maximum growth rate for these Q will probably lie somewhere near T = 1/2. The situation is only slightly different for other b. As b increases we find that the region of instability extends to somewhat smaller values of y except for the narrow tail at large -- Q which still persists but which exhibits smaller growth rates.
On open ocean baroclinic instability in the Arctic One can also inquire of this model what the effect of friction against the ice will be on the perturbations. We do not ask for its effect on the basic state, a question which involves the details of the driving mechanisms. If friction is represented by a constant eddy viscosity Ekman layer near z = 0, the upper perturbation boundary condition becomes
(1 - c)
d9= (i
iT2E1/~b'~
I%
so that friction is incorporated into the coefficient H=
Ea/~b m
Ro~t
b~,* v e / ( f ) 2riD U
which may be interpreted as the inverse boundary layer Reynolds number weighted by the aspect ratio of the perturbation. Using numbers obtained from the inviscid theory to guess L* ~ 105 m for the Arctic (see below), with U = 0.1 m s-a, D -- 3000 m, vE = 0.01 m s s-1 corresponding to an Ekman e-folding depth of 10 m, we find that H is about 1. Figure 2 shows contours of cI and c, for H = 0.5, b = 6. The presence of friction causes
generally smaller growth rates and an apparent damping of instabilities which causes modes with significant ci's to occur at small values of y, presumably a consequence of the T2 factor multiplying the frictional terms. At large H > 4 the only significant modes remaining occur for Q near - 1 , 0 < y < 1/2. We have found that baroclinic instability of the deep ocean with exponential profiles of p(z), Uo(z ) will only occur for T ~ 1, regardless of topography and friction within reasonable ranges. Note that for any slope, Q can be made to equal --1 by an appropriate choice of A. However, this puts a further limitation on ~ since as T is fixed goes like (1 + tan2A) -a < 1. This will lengthen the perturbation scales and decrease the dimensional growth rate ~. = 2 n U q
The wavelength ~,* is given by
/r
2,DN/L(l + t a n " A ) g
-I0 -
~-~ oPod
/ -.05
0
641
0
-I.0
I
0
.5
1.0
Fig, 2 (a)
1.5
I
2.0
642
J . E . HART and P. D.
-I0
-
KILLWORTH
./
.i
d Q -to
-~1 0
I .5
] 1.0
I 1.5
I 2.0
Fig. 2 (b) Fig. 2. Contours of ci (a) and c, (b) for b = 6, H = 0.5. We see that eddy sizes scale with the radius of deformation based on the surface stratification and total depth. This is because the lower boundary is necessary for instability when [~ = 0. If the lower boundary condition is replaced by a condition that q~ = 0 or q~z : 0 say, the integral theorems of PEOLOS~Y (1964) show that no instability is possible. If we use numbers appropriate for the interior Arctic where D ~ 3000 m, N(0) ~ 0.01, with b : 6.28 we find that )~* 3> 600 kin, or scales ~.*/2 no smaller than 300 km. Since the measurements of N~wxoN, AAGAARD and COACHMAN (1974) and HUNKINS (1974) show wavelengths considerably shorter than this, we conclude that open ocean instability is not a likely candidate for the production of these eddies. Also note that this 3~* is a substantial fraction of the basin scale so it is probable that curvature effects may be important. With this L* and c i ~ 0.2, ~* ~ l0 -7 s-t. The growth time is greater than 140 days. This instability may be related to observed long-period, larger-scale fluctuations (see below). Since large critical wavelengths were found in the preceding, we expect that the inclusion of planetary vorticity advection may change the results somewhat. To check this we have solved the perturbation problem numerically (cf. Fig. 3).
The basic state has the same shape as above and is directed westward with a peak speed of 10 cm s-1. The variation of f with latitude has the value appropriate for 70°N. For a depth of 3000 m the growth rate ~*ci* peaks at ~* ~ 2 x 10-7 cm --1 (L* ~ 300 kin) with a value of the order of 2 × ]0 -7 s -1. As opposed to the case without [~, there appears to be no shortwave cutoff. Further, if the baroclinic flow is eastward, the growth rates become O(10- 9 s-l), which follows from a result of GILL, GREEN and SIMMONS (1974). Thus, the direction of the baroclinic flow is relevant when the [~-effect is retained. However, no matter what the direction of the basic flow, the growth-rates are negligibly small (of order 10-s s-1) at wavelengths of order 10 kin.
5. CONCLUSIONS We have shown that the mean circulation in the Canadian basin is, for eddies of the observed size, essentially baroclinically stable. It is suggested that if baroclinic instability is the cause of the observed short wavelength eddies, it either must occur in shallower water where the shears are larger and extend to the bottom, or perhaps it may occur on open ocean time-dependent 0 6 to 32 weeks), large-scale (100 to 200 km), high-
On open ocean baroclinic instability in the Arctic
643
a , .~ N - ~ U 5D With D ~ 200 m and U ~ 0.1 m s-1, this leads to growth times on the order of a few days. Figure 4 shows results of some numerical computations on a B-plane with D -----200 m,
c; -5
Uo* = 0.1 m s-1 • (z* + D)/D -I0
5x1(37
a° C/
o
I
0
5 x l O "r
I0 -6
a I' (cm-1}
Fig. 3. Curves of growth rate tt*c**(s-t) and phase speed c,* (em sJ ) for westward shear flow U0* ~ 0.1 m s-x ~*, in depth D = 3000 m over a N-S slope at 70°N. Curve (a) is for a fiat bottom, (b) bottom slope OD/dy = 2.7 × 10-*, (c) bottom slope -- 2.7 x 10-*. The growth rates are extremely small if the shear becomes eastward.
shear flows such as those observed by NEWTON (1973). The first alternative can be qualitatively assessed by referring to the f-plane Eady-problem of a uniformly sheared horizontal current over a slope discussed by BLtrMSACKand GmRASCH (1972) a m o n g others. The shear extends to the bottom although the basic velocity is presumed to be zero there. The perturbation length scale is X* -- 6ND
f~ where ~t for the fastest growing wave depends on the ratio o f the bottom slope* to isopyenal slope. I f this is negative, instability with reasonable growth rates can occur for ~t ~ 4. Thus in shallow water (D ~ 200 m), wave scales on the order o f 10 km can be obtained. The growth rate now scales like
directed westward at 70°N over a north-south bottom slope. The presence of the negative slope shifts the instability to shorter wavelengths while decreasing the growth-rate slightly. Again e-folding times of a few days are achieved at wave scales of about 15 km. For such wavelengths, the B-effect becomes small, so that reversing the direction of the baroclinic flow is only quantitatively significant in the sense that the sign of the isopycnal/topographic slope ratio changes. However, this possibility tacitly assumes that the eddies, once formed on the continental shelf, are able to leave it and propagate into the open ocean. I f it is assumed that open-ocean disturbances propagate as linear normal modes, then each normal mode is triggered in some degree by the shelf disturbance. We briefly consider three possibilities. First, the open ocean may respond to the frequency of the instability (about 5 x 10-e s-X). Consideration of the dispersion relation for each mode shows that all but the barotropic mode are trapped near the shelf; and the barotropic mode itself would be associated with wavelengths considerably larger than the scale of the Arctic Ocean. Second, if the ocean responds to the wavelength of the instability (about 30 km), frequencies in the interior of 7 x 10-s s-1 and lower would be triggered. Such periods are too long to be acceptable. The third possibility is that the ocean responds to the vertical shape of the perturbation velocity (confined to the top 200 m). Straightforward calculation shows that most energy goes into the first and second baroclinic *Slopes where the parameter (slope) x LSf/DU is not too large.
644
J . E . HART and P. D. KILLWORTH
-3 -4 o
Cr -5 / I -6
3xlO-6 2xlO 6
c3.° C i" 10-6
0 0
5x10-7
I0 -6
1.5× I
~2" (c m-.I )
Fig. 4. Curves of growth rate ct*ci* (s 1) and phase speed c,* (cm s -~) for westward shear flow Uo* -- 0.1 m s x (z* + D)/D, in depth D = 200 m water over a N-S slope at 70°N. Curve (a) is for a fiat bottom, (b) bottom slope OD/Oy = 6.8 × 10-5, (c) bottom slope --6.8 :,~ 10-'%
modes. However, these are typically the modes which are unstable in the open ocean, giving problems in interpretation (KILLWORTH and ANDERSON, 1975a). The fact that the eddies are significantly non-linear may lead to other, more complicated, propagation or advection effects which need further study. The second alternative is more difficult to assess. The stability o f a baroclinic topographic/ planetary wave should be considered in detail. Preliminary calculations on the stability o f an oscillatory zonal current U0 = e bz (1 + A cos cot) indicate that substantially higher open ocean growth rates can be obtained during the high-shear part of the cycle than those computed in Section 4 using a steady basic current (KILLWORTH and ANDERSON, 1975b). However, since this presumed form for U0 does not satisfy the basic equations of motion the results are questionable. The instability problem for wavy basic motions needs further study, especially in regard to finite-amplitude
equilibration or saturation since the small-scale eddies have velocities several times larger than those o f the medium-scale waves on which they are supposed to grow. Although there is sufficient potential energy at medium scale to account for the small-scale kinetic energy, it is not k n o w n if the transformation is dynamically permissible. The problem o f the eddy kinetic energy may not arise if they are generated off the north coast o f Alaska or near the Bering inflow since the currents there are larger.
Acknowledgments -~The work of J. E. Hart was supported by the National Science Foundation under grant DES 74-14356; that of P. D. Killworth by the Natural Environment Research Council. REFERENCES
BLUMSACKS. L. and P. J. GmRASCH(1972) Mars: the effects of topography on baroclinic instability. Journal of the Atmospheric Sciences, 29, 1081-1089. GILL A. E., J. S. A. GREEN and A. J. SIMMONS(1974)
On open ocean baroclinic instability in the Arctic Energy partition in the large-scale ocean circulation and the production of mid-ocean eddies. Deep-Sea Research, 21, 499-528. HtJNKINS K. L. (1974) Subsurface eddies in the Arctic Ocean. Deep-Sea Research, 21, 1017-1034. KmLWORTH P. D. and D. L. T. ANOERSON(1975a) Meaningless Modes ? MODE Hot-line News, 71. (Unpublished document.) KmLWORrH P. D. and D. L. T. ANDERSOr~(1975b) Some thoughts on eddy generation. MODE Hotline News, 72. (Unpublished document.) NEWTONJ. L. (1973) The Canada Basin, mean circula-
645
tion and intermediate scale flow features. Ph.D. Thesis, University of Washington, 158 pp. N~WTON J. L., K. AAGAARDand L. K. COACHMAN (1974) Baroclinic eddies in the Arctic Ocean. Deep-Sea Research, 21, 707-721. P~DLOSKV J. (1964) The stability of currents in the atmosphere and the ocean. Journal of the Atmospheric Sciences, 21, 201-219. ROaINSON A. R. and J. C. McWILLL~MS(1974) The baroclinic instability of the open ocean. Journal of Physical Oceanography, 4, 281-294.
On open ocean baroclinic instability in the Arctic J. E. HART* and P. D. K1LLWORTHt (Received from first author 12 March 1975; revision receivedfrom both authors July i 975) simple linear model is used to investigate the possibility of baroclinic instability in the Arctic. The basic state consists of a barotropic flow directed along depth contours, a baroclinic component uniform in horizontal directions which decreases exponentially with depth, and an exponentially decaying stratification. With no ~-effect there is always a shortwave cutoff for wavelengths shorter than the radius of deformation based on the surface N ~and total depth. With [3there appears to be no shortwave cutoff but the growth rate peaks near the same values of wavelength. The results suggest that the 10- to 20-km eddies observed in the Arctic are probably not generated by baroclinic instability in the open ocean but may arise from instability in the shallow regions. Abstract--A
1. I N T R O D U C T I O N
RECENT observations of NEWTON, AAGAARD a n d COACHMAN(1974) and HUNKINS(1974) have shown the presence of substantial eddy activity in the interior of the Canada Basin of the Arctic Ocean. The eddies have horizontal scales from 10 to 20 km and velocities on the order of 30 cm s-1. We investigate here the hypothesis that the eddies are the result of a baroclinic instability process in the open ocean, or in shallow water near continental margins. Open ocean baroclinic instability would appear a priori to be a likely candidate for the observed eddies. This is because the observed eddy sizes agree well either with the first-mode Rossby radius of deformation calculated from the observed stratification (25 km), or with a radius of deformation based on the depth over which the stratification is important, which is, as Hunkins shows, about 17 km. However, both the [3-effect and the bottom topography can be essential elements of the instability process. The theory developed in Sections 2 to 4 shows that the unstable wavelengths, for realistic stratification and mean currents, scale with the parameter N(0) D where
f
unstable waves in the open ocean whose length scales are 10 to 20 times larger than the radius of deformation. It is thus concluded (Section 5) that if baroclinic instability produces the observed eddies, the instability must occur in the shallower parts of the Arctic Ocean where large shears can extend to the bottom. 2. THE QUASI-GEOSTROPHIC EQUATION
Most of the analysis in this paper concerns the stability of an inviscid quasi-geostrophic flow on an f-plane. For small wavelengths, order 20 km, the neglect of the [~-effectis apparently permissible; however, for wavelengths of order 100 km or more, [~ should be included, and in Section 4 the stability problem will be solved numerically including planetary vorticity advection. The governing equation is the familiar potential vorticity equation (with [3 retained for clarity, although it will often be set to zero):
d [ v n ' , + O ~c3hb+ [3y] dt
O,
(2.1)
Oz c3z
with d d-t ~ O-t + J(q;'
)"
(2.2)
N(0) is the Brunt-Viiis/il~i frequency just below *Department of Meteorology, Room 54-1713, Massathe mixed layer, f the Coriolis parameter, and D the total depth. (In fact, the radius of deformation chusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A. is also proportional to this factor, but with a -[-Department of Applied Mathematics and Theoretical numerical factor of about one-tenth.) This yields Physics, Cambridge University, Cambridge, England. 637
638
J.E. HARTand P. D. KILLWORTH
Asterisked variables are dimensional; nonasterisked variables are dimensionless. The flow is geostrophic, so that
where q~is the dynamic part of the pressure, scaled by p o f L U , and p is the deviation from the background density 15(z), normalized by P0. Velocities have been scaled by U, the scale of the mean flow whose stability is to be tested, and horizontal scales (x* east, y* north) by L. Depths (z* ; z* =- 0 at the surface) are scaled by D and time by L/U. Finally, ~ == ~*L2/U. When ~ ~ 0 the only parameter appearing in the interior equation is
f2L2po -- gD2-iOP-~-Z* i
"
(2.3)
The boundary conditions are that d (e+~ + ha ) dt
0
(2.4)
fL
+o = F(hB) + "~(x,y,z). To simplify the analysis we assume that hB contours are locally unidirectional so that the bottom slope is uniform over a typical wave scale and •qb 0 is effectively constant in x and y. This type of approximation was derived by a formal perturbation expansion for the two-layer model by ROmNSON and MCWlLLIAMS (1974). For arbitrary orientation of the horizontally uniform baroclinic basic flow component to the bottom contours, the interior equation (1.1) is satisfied if we choose ~ --ye b~. Thus the baroclinic component of the mean current is oriented in the x direction. We now perturb about this basic state, writing ~ ' ---- ,~?(z) e i("x~ 8y--
at z ---- 0 and z =~ --1. In this equation h B --= 0 at z ----0 and hs = h*/DRo at z = --1. Here h* is the height of the bottom topography relative to z* ---- - - D , and R 0 is the Rossby number U R o ~- ~ .
where b is determined by depth and generally takes a value of about 6 to 8. The stratification then decays to 2 to 10% of its surface value in about 1000 m. The basic velocity distribution is made up of both barotropic and baroclinic parts. The barotropic part is constrained to be parallel to the depth contours, so the basic state may be written
(2.5)
The perturbation makes an angle A : tan-l~/~ with the baroclinic basic flow Uo -- e b~. It can easily be shown that the presence of a uniform barotropic current Uo only affects the real part o f . , the frequency of the perturbation, not its growth or decay. If we write
= ~Ui,cos 0 + ~Ubsin 0 + 7c, These equations have been used by PEDLOSKY (1964) and many others in similar studies of where 0 is the angle between Uo and the depth baroclinic instability. contours, all reference to the barotropic flow is removed. The linearization of (2.1) and (2.4) about 3. THE BASIC STATE AND PERTURBATION ~b0, using the presumed forms for ~b' and ~ yields EQUATIONS a two-point boundary value problem for the A good representation of the stable density determination of ~c, the perturbation growth rate. stratification of the interior Arctic is achieved by A uniform topographic slope is included for writing generality. It is represented in the parameter S ---- ~h*/DR o in which ~h* is the upslope change e0 e0 of depth over a length L, D is the average depth, --~ z* = 0 . e , and R o is the Rossby number U/fL. With some ~z* ~z* [ b~
On open ocean baroclinic instability in the Arctic straightforward manipulation and transformation to the logarithmic coordinate rl = d '~, the stability problem for [~ : 0 can be reduced to
dq~ (1 - - c ) ~ = q ~
--c~
d~p
in effect the radius of deformation based on surface stratification and D. The general solution of (3.1) is q~ = 111/211 (2Tr111~) + B
d29
+ Qq~ = 0
atrl = 1
(3.2)
atrl --- 0.
(3.3)
639
TII'~
K1 (2T~1/~),
(3.7)
where the coefficient of the first term is set equal to 1 as a normalization. Using (3.2) and (3.3) and the recursion relations for modified Bessel functions we obtain an eigenvalue equation for c, c~T~ [K0(27) + I0(2T)M] + cy [--yK2(2y) -- M712(22') -- QIo(2y)/2V]
The parameters entering this eigenvalue problem are a non-dimensional wavenumber scaled on the surface deformation radius,
+ Q/~(2y) = 0,
(3.8)
where M ~- In (2?) -- b/2. ~ / =
b N / ( 1" + X'~ n ' A ")
,
(3.4)
where
4. RESULTS At large enough Y so that Ko,e(2y) ~ Io,,(2Y) the discriminant can be written as
= . f~L2P0
gD l°o l I&z* I o
(3.5)
and S cos 0 Q---(1 b~
+ tan 0 tan A) -- 1. (3.6)
In writing (3.3) we have assumed that Uo(z) = 0 at z = -- 1. The error in doing this is e-b which means that we may be omitting the possibility of unstable modes whose growth rates would be of order e -b relative to the growth rates found below. Note that the key parameters y and Q are independent of the scaling length L as is the dimensional growth rate **. L enters as the horizontal scale of the variations of the basic state. For our local analysis to be valid we must have = 2~tL/X* be large. However, the critical parameter in the stability analysis is y which is essentially a measure of the wave scale k* in units of 2~tL ~,bV'~ '
T'[MI2(27) -- Qlo(2y)/272] 2. Thus, at large ? (2 or 3 may be large enough because of the factor 2 in the argument) the flow will be stable. Thus this exponentially stratified and sheared flow exhibits a short wave cutoff. For small y the discriminant becomes 74[(1 + Q ) Z (1 + Q ) ( l n 2 y - ~) 4~,~ + + ~Q- + 0(74)] ~ Thus as ? l+ 0 there will be a long wave cutoff at finite T unless Q = - 1 . Finally, note that since there is no instability at large y, the quantity M is negative in that range of T for which instability is possible.'f Using the asymptotics of K0 and I o it is possible to show that for our values of b, unstable modes can only be found for negative Q. Thus, there would appear to be a fairly small range in ~fWe assume b is _> 4.
640
J.E.
HART a n d P. D . KaLLWORWrt
-10 -
0
Q
-I.0
-.I
0
.5
I
L
J
1.0
k5
20
Fig. 1 (a) -I0 -
(~ -L0
2 ~
©
59 . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
1.0
] .................
15
2.0
Fig. 1 (b) Fig. 1.
C o n t o u r s o f c~ (a) a n d c, (b) for b
parameter space where instability is possible. Figure 1 shows isolines o f c i and c, as functions o f Q and y for b = 6. The general asymptotic results are reflected in the c I diagram. It is seen that the largest ci occur near 7 = 1/2 for Q = - 1 . For Q near - 1 the largest c~ occur for small V but recall that the actual growth rate is :{ci so that as
6, H = 0 (inviscid).
T oc :{ the maximum growth rate for these Q will probably lie somewhere near T = 1/2. The situation is only slightly different for other b. As b increases we find that the region of instability extends to somewhat smaller values of y except for the narrow tail at large -- Q which still persists but which exhibits smaller growth rates.
On open ocean baroclinic instability in the Arctic One can also inquire of this model what the effect of friction against the ice will be on the perturbations. We do not ask for its effect on the basic state, a question which involves the details of the driving mechanisms. If friction is represented by a constant eddy viscosity Ekman layer near z = 0, the upper perturbation boundary condition becomes
(1 - c)
d9= (i
iT2E1/~b'~
I%
so that friction is incorporated into the coefficient H=
Ea/~b m
Ro~t
b~,* v e / ( f ) 2riD U
which may be interpreted as the inverse boundary layer Reynolds number weighted by the aspect ratio of the perturbation. Using numbers obtained from the inviscid theory to guess L* ~ 105 m for the Arctic (see below), with U = 0.1 m s-a, D -- 3000 m, vE = 0.01 m s s-1 corresponding to an Ekman e-folding depth of 10 m, we find that H is about 1. Figure 2 shows contours of cI and c, for H = 0.5, b = 6. The presence of friction causes
generally smaller growth rates and an apparent damping of instabilities which causes modes with significant ci's to occur at small values of y, presumably a consequence of the T2 factor multiplying the frictional terms. At large H > 4 the only significant modes remaining occur for Q near - 1 , 0 < y < 1/2. We have found that baroclinic instability of the deep ocean with exponential profiles of p(z), Uo(z ) will only occur for T ~ 1, regardless of topography and friction within reasonable ranges. Note that for any slope, Q can be made to equal --1 by an appropriate choice of A. However, this puts a further limitation on ~ since as T is fixed goes like (1 + tan2A) -a < 1. This will lengthen the perturbation scales and decrease the dimensional growth rate ~. = 2 n U q
The wavelength ~,* is given by
/r
2,DN/L(l + t a n " A ) g
-I0 -
~-~ oPod
/ -.05
0
641
0
-I.0
I
0
.5
1.0
Fig, 2 (a)
1.5
I
2.0
642
J . E . HART and P. D.
-I0
-
KILLWORTH
./
.i
d Q -to
-~1 0
I .5
] 1.0
I 1.5
I 2.0
Fig. 2 (b) Fig. 2. Contours of ci (a) and c, (b) for b = 6, H = 0.5. We see that eddy sizes scale with the radius of deformation based on the surface stratification and total depth. This is because the lower boundary is necessary for instability when [~ = 0. If the lower boundary condition is replaced by a condition that q~ = 0 or q~z : 0 say, the integral theorems of PEOLOS~Y (1964) show that no instability is possible. If we use numbers appropriate for the interior Arctic where D ~ 3000 m, N(0) ~ 0.01, with b : 6.28 we find that )~* 3> 600 kin, or scales ~.*/2 no smaller than 300 km. Since the measurements of N~wxoN, AAGAARD and COACHMAN (1974) and HUNKINS (1974) show wavelengths considerably shorter than this, we conclude that open ocean instability is not a likely candidate for the production of these eddies. Also note that this 3~* is a substantial fraction of the basin scale so it is probable that curvature effects may be important. With this L* and c i ~ 0.2, ~* ~ l0 -7 s-t. The growth time is greater than 140 days. This instability may be related to observed long-period, larger-scale fluctuations (see below). Since large critical wavelengths were found in the preceding, we expect that the inclusion of planetary vorticity advection may change the results somewhat. To check this we have solved the perturbation problem numerically (cf. Fig. 3).
The basic state has the same shape as above and is directed westward with a peak speed of 10 cm s-1. The variation of f with latitude has the value appropriate for 70°N. For a depth of 3000 m the growth rate ~*ci* peaks at ~* ~ 2 x 10-7 cm --1 (L* ~ 300 kin) with a value of the order of 2 × ]0 -7 s -1. As opposed to the case without [~, there appears to be no shortwave cutoff. Further, if the baroclinic flow is eastward, the growth rates become O(10- 9 s-l), which follows from a result of GILL, GREEN and SIMMONS (1974). Thus, the direction of the baroclinic flow is relevant when the [~-effect is retained. However, no matter what the direction of the basic flow, the growth-rates are negligibly small (of order 10-s s-1) at wavelengths of order 10 kin.
5. CONCLUSIONS We have shown that the mean circulation in the Canadian basin is, for eddies of the observed size, essentially baroclinically stable. It is suggested that if baroclinic instability is the cause of the observed short wavelength eddies, it either must occur in shallower water where the shears are larger and extend to the bottom, or perhaps it may occur on open ocean time-dependent 0 6 to 32 weeks), large-scale (100 to 200 km), high-
On open ocean baroclinic instability in the Arctic
643
a , .~ N - ~ U 5D With D ~ 200 m and U ~ 0.1 m s-1, this leads to growth times on the order of a few days. Figure 4 shows results of some numerical computations on a B-plane with D -----200 m,
c; -5
Uo* = 0.1 m s-1 • (z* + D)/D -I0
5x1(37
a° C/
o
I
0
5 x l O "r
I0 -6
a I' (cm-1}
Fig. 3. Curves of growth rate tt*c**(s-t) and phase speed c,* (em sJ ) for westward shear flow U0* ~ 0.1 m s-x ~*, in depth D = 3000 m over a N-S slope at 70°N. Curve (a) is for a fiat bottom, (b) bottom slope OD/dy = 2.7 × 10-*, (c) bottom slope -- 2.7 x 10-*. The growth rates are extremely small if the shear becomes eastward.
shear flows such as those observed by NEWTON (1973). The first alternative can be qualitatively assessed by referring to the f-plane Eady-problem of a uniformly sheared horizontal current over a slope discussed by BLtrMSACKand GmRASCH (1972) a m o n g others. The shear extends to the bottom although the basic velocity is presumed to be zero there. The perturbation length scale is X* -- 6ND
f~ where ~t for the fastest growing wave depends on the ratio o f the bottom slope* to isopyenal slope. I f this is negative, instability with reasonable growth rates can occur for ~t ~ 4. Thus in shallow water (D ~ 200 m), wave scales on the order o f 10 km can be obtained. The growth rate now scales like
directed westward at 70°N over a north-south bottom slope. The presence of the negative slope shifts the instability to shorter wavelengths while decreasing the growth-rate slightly. Again e-folding times of a few days are achieved at wave scales of about 15 km. For such wavelengths, the B-effect becomes small, so that reversing the direction of the baroclinic flow is only quantitatively significant in the sense that the sign of the isopycnal/topographic slope ratio changes. However, this possibility tacitly assumes that the eddies, once formed on the continental shelf, are able to leave it and propagate into the open ocean. I f it is assumed that open-ocean disturbances propagate as linear normal modes, then each normal mode is triggered in some degree by the shelf disturbance. We briefly consider three possibilities. First, the open ocean may respond to the frequency of the instability (about 5 x 10-e s-X). Consideration of the dispersion relation for each mode shows that all but the barotropic mode are trapped near the shelf; and the barotropic mode itself would be associated with wavelengths considerably larger than the scale of the Arctic Ocean. Second, if the ocean responds to the wavelength of the instability (about 30 km), frequencies in the interior of 7 x 10-s s-1 and lower would be triggered. Such periods are too long to be acceptable. The third possibility is that the ocean responds to the vertical shape of the perturbation velocity (confined to the top 200 m). Straightforward calculation shows that most energy goes into the first and second baroclinic *Slopes where the parameter (slope) x LSf/DU is not too large.
644
J . E . HART and P. D. KILLWORTH
-3 -4 o
Cr -5 / I -6
3xlO-6 2xlO 6
c3.° C i" 10-6
0 0
5x10-7
I0 -6
1.5× I
~2" (c m-.I )
Fig. 4. Curves of growth rate ct*ci* (s 1) and phase speed c,* (cm s -~) for westward shear flow Uo* -- 0.1 m s x (z* + D)/D, in depth D = 200 m water over a N-S slope at 70°N. Curve (a) is for a fiat bottom, (b) bottom slope OD/Oy = 6.8 × 10-5, (c) bottom slope --6.8 :,~ 10-'%
modes. However, these are typically the modes which are unstable in the open ocean, giving problems in interpretation (KILLWORTH and ANDERSON, 1975a). The fact that the eddies are significantly non-linear may lead to other, more complicated, propagation or advection effects which need further study. The second alternative is more difficult to assess. The stability o f a baroclinic topographic/ planetary wave should be considered in detail. Preliminary calculations on the stability o f an oscillatory zonal current U0 = e bz (1 + A cos cot) indicate that substantially higher open ocean growth rates can be obtained during the high-shear part of the cycle than those computed in Section 4 using a steady basic current (KILLWORTH and ANDERSON, 1975b). However, since this presumed form for U0 does not satisfy the basic equations of motion the results are questionable. The instability problem for wavy basic motions needs further study, especially in regard to finite-amplitude
equilibration or saturation since the small-scale eddies have velocities several times larger than those o f the medium-scale waves on which they are supposed to grow. Although there is sufficient potential energy at medium scale to account for the small-scale kinetic energy, it is not k n o w n if the transformation is dynamically permissible. The problem o f the eddy kinetic energy may not arise if they are generated off the north coast o f Alaska or near the Bering inflow since the currents there are larger.
Acknowledgments -~The work of J. E. Hart was supported by the National Science Foundation under grant DES 74-14356; that of P. D. Killworth by the Natural Environment Research Council. REFERENCES
BLUMSACKS. L. and P. J. GmRASCH(1972) Mars: the effects of topography on baroclinic instability. Journal of the Atmospheric Sciences, 29, 1081-1089. GILL A. E., J. S. A. GREEN and A. J. SIMMONS(1974)
On open ocean baroclinic instability in the Arctic Energy partition in the large-scale ocean circulation and the production of mid-ocean eddies. Deep-Sea Research, 21, 499-528. HtJNKINS K. L. (1974) Subsurface eddies in the Arctic Ocean. Deep-Sea Research, 21, 1017-1034. KmLWORTH P. D. and D. L. T. ANOERSON(1975a) Meaningless Modes ? MODE Hot-line News, 71. (Unpublished document.) KmLWORrH P. D. and D. L. T. ANDERSOr~(1975b) Some thoughts on eddy generation. MODE Hotline News, 72. (Unpublished document.) NEWTONJ. L. (1973) The Canada Basin, mean circula-
645
tion and intermediate scale flow features. Ph.D. Thesis, University of Washington, 158 pp. N~WTON J. L., K. AAGAARDand L. K. COACHMAN (1974) Baroclinic eddies in the Arctic Ocean. Deep-Sea Research, 21, 707-721. P~DLOSKV J. (1964) The stability of currents in the atmosphere and the ocean. Journal of the Atmospheric Sciences, 21, 201-219. ROaINSON A. R. and J. C. McWILLL~MS(1974) The baroclinic instability of the open ocean. Journal of Physical Oceanography, 4, 281-294.