Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ocean.phys.msu.ru/courses/stathyd/2002%20Smyth%20et%20al.%2C%20Shear%20instability%20and%20gravity%20wave%20saturation%20in%20an%20asymmetrically%20stratified%20jet.pdf Äàòà èçìåíåíèÿ: Wed Nov 11 15:01:13 2009 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:56:32 2012 Êîäèðîâêà:

Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

Shear instability and gravity wave saturation in an asymmetrically stratified jet
W.D. Smyth and J.N. Moum College of Oceanic and Atmospheric Sciences Oregon State University, Corvallis OR 97331-5503 Submitted July 17th, 2001.


Corresponding author. Telephone: (541) 737-3029; graphic gravity waves in the middle atmosphere. Transemail: smyth@coas.oregonstate.edu mission beyond the lower flank may partly account for wave signals observed in the deep equatorial ocean. All of these processes exert zonal forces on the jet that alter its speed and shape. The wave structures and assoAbstract ciated fluxes developed by the idealized model are comMotivated by the mean current and stratification structure pared with observations of the EUC. Keywords: associated with the Equatorial Undercurrent (EUC), we instability, internal waves, jets, stratified shear flow. examine the stability and wave propagation characteristics of a highly idealized model flow: the asymmetrically stratified jet. This is a parallel shear flow in which the depth-varying current has the sech2 form of a Bickley jet. The stratification has a step function structure: the buoyancy frequency takes uniform values above and below the center of the jet, with the larger value occurring below. The spectrum contains three classes of unstable normal modes. Two are extensions of the sinuous and varicose modes of the unstratified Bickley jet; the third has not been described previously. The asymmetric stratification structure allows instabilities to radiate gravity wave energy from the upper flank of the jet to the lower flank, where it encounters a critical layer. From here, wave energy may be reflected, absorbed or transmitted. Absorption results in wave saturation and momentum transfer to the mean flow, in close analogy with the breaking of oro1

1

Introduction

A dominant component of the tropical ocean current system is the equatorial undercurrent (EUC), a region of intense turbulence associated with shear instability and internal gravity wave radiation. Our purpose here is to explore some of the fundamental properties of instability and wave radiation in this regime via stability analysis and numerical simulation of a simple model flow. The EUC flows eastward at a depth of approximately 100m, spans a meridional range of a few hundred km centered on the equator, and has a maximum speed near 1m/s. The current is a nearly permanent feature in both Atlantic and Pacific oceans, and is forced by the zonal pressure


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

gradient that builds up due to westward transport of surface water by the Trade Winds (e.g. Veronis (1960), see Qiao and Weisberg (1997) for a recent discussion). Interannual fluctuations in the strength of the Pacific EUC are an integral component of the El Nino - Southern Oscillation (ENSO) cycle. Shear on the upper and lower flanks of the EUC drives much of the strong turbulence that characterizes the equatorial oceans (Moum et al., 1989; Peters et al., 1989). The resulting vertical momentum flux distributes momentum input from the wind into the upper ocean and thereby provides a partial counterbalance to the zonal pressure gradient (Hebert et al., 1991). The EUC is also a region of strong internal gravity wave (IGW) activity, which may also account for significant momentum fluxes (Moum et al., 1989; Lien et al., 1996). Dillon et al. (1989) found that momentum fluxes due to turbulence act over a relatively shallow layer adjacent to the surface (i.e. much shallower than the layer over which the zonal pressure gradient acts), and therefore suggested that fluxes due to IGW are also needed to explain the observed current structure. Application of normal mode stability analysis to the EUC began with Sutherland (1996), who demonstrated that the upper flank of the EUC may support shear instabilities that act to radiate energy downward in the form of IGW. Mack and Hebert (1997) and Sun et al. (1998) showed that the dominant IGW signal observed in the equatorial Pacific is associated with this class of instabilities. Using observed profiles of velocity and density, Sun et al. (1998) demonstrated that shear instability can flux momentum to the deepest reaches of the EUC, in accordance with the suggestion of Dillon et al. (1989) discussed above. In order that these results be applicable under a range of equatorial flow regimes, the mechanisms of instability must be understood more thoroughly. To this end, we are analyzing a sequence of simple models that converge towards an accurate representation of the EUC and 2

whose instability mechanisms are both broadly relevant and accessible to detailed study (as opposed to the highly complex and flow-specific instabilities of measured currents, e.g. Sun et al. (1998)). Our hypothesis is that the EUC exists, on average, in a nearly stable state, but that this state is frequently perturbed by fluctuations in forcing. The flow adjusts to these perturbations via a range of instability mechanisms that lead to property fluxes via waves and turbulence. We begin here with a model that isolates the dominant features of the shear and stratification that characterize the EUC. The stability characteristics of the EUC are governed in large part by a distribution of stable density stratification that is asymmetric with respect to the current. In particular, stratification tends to be weak on the upper flank of the current, which includes the wind-driven South Equatorial Current and the surface mixed layer (figure 1a,b). Because of this asymmetry, typical values of the gradient Richardson number tend to be smaller on the upper flank (often close to or less than the critical value 1/4) and larger on the lower flank (figure 1c). Unstable modes of a jetlike current such as the EUC each possess a pair of critical levels (levels at which the mean flow velocity matches the phase velocity of the mode), one on each flank of the jet. A wave energy flux incident on a critical level interacts with the mean flow, with the result that it is divided into reflected, absorbed, and transmitted components. The details of this division depend largely on the local value of the gradient Richardson number, Ri. If Ri < 1/4 at the critical level, the wave may be over-reflected, i.e. it may gain energy from the mean flow during the interaction, so that the sum of the reflected and transmitted components is greater than the incoming flux. This process is thought to be a central aspect of shear instability (e.g. Lindzen and Tung, 1978; Smyth and Peltier, 1989). If Ri > 1/4 at the critical level, over-reflection cannot occur, and waves lose energy to the mean flow. In this case, trapping of wave energy near the


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

critical layer leads to turbulent breakdown via convective instability (Winters and D'Asaro, 1993; Lin et al., 1993; Winters and D'Asaro, 1994). The asymmetric distribution of stratification in the EUC raises the possibility of modes that have local Richardson number less than 1/4 at the upper critical level but greater than 1/4 at the lower critical level. In such a case, we expect that wave energy will be radiated from the upper critical level downward through the jet core and to the lower critical level. Some of this energy will then be reflected back to the upper critical level, sustaining the instability. Some of the energy may be transmitted through the lower critical level, possibly propagating to great depths. Finally, some energy may be absorbed at the lower critical level, causing momentum transfer and turbulent mixing. The pattern of turbulence observed in the EUC is suggestive of the processes described above, with enhanced turbulence occurring on the lower flank of the jet where the Richardson number is relatively high (figure 1d, z < -160m). This mechanism is analogous to the saturation of orographic gravity waves in the middle atmosphere. Parameterization of the momentum flux divergences generated by that wave-mean flow interaction has turned out to be crucial to the accuracy of atmospheric general circulation models (e.g. Lindzen, 1981; Fritts, 1989). Here, we investigate instability and wave radiation in the EUC using the simplest possible model, the asymmetrically stratified jet (figure 2). The undercurrent is represented by a sech2 velocity profile, i.e. a Bickley jet (Drazin and Reid, 1981). The buoyancy frequency is described by a step function whose value changes at the center of the jet (see section 2 for details). This model is relevant to any jetlike current that exists far from boundaries in the presence of asymmetric stratification, atmospheric jet streams being another example. To correspond with the EUC, we let the larger value of the buoyancy frequency occur in the lower half-space, but the properties of 3

0

(a)
-20 -40 -60 -80

(b)

(c)

(d)

z (m)

-100 -120 -140 -160 -180 -200

0

0.5

1

0

5

U (m/s)

N2 (10-4s-2)

10

-1

10

0

10

1

10

-9

10

-8

Ri

(m /s )

10 10 23

-7

-6

Figure 1: Observations taken aboard the R/V Wecoma at 0o N ,
140o W , during the Tropical Instability Wave Experiment (Lien et al., 1995). Profiles are averaged over year days 311-315, inclusive, of 1991. (a) Zonal velocity, (b) squared buoyancy frequency, (c) gradient Richardson number (the vertical line marks the critical value 1/4), (d) kinetic energy dissipation rate.

a jet with stratification larger above its core are obtained trivially from the present results. Instabilities of the stratified Bickley jet were first studied numerically by Hazel (1972). This work was continued by Sutherland and Peltier (1992) and Sutherland et al. (1994), who added a region of uniform flow to the center of the jet. Here, we extend these investigations by allowing the stratification to be asymmetric with respect to the current. Sutherland (1996) studied waves and instability specific to the EUC using a model that included the boundary at the surface, but which represented the mean current as a hyperbolic tangent shear layer, thus isolating the upper flank the EUC. He allowed stratification to be weaker above the shear layer than below, and showed that this promotes the growth of unstable modes that couple efficiently with the deeper IGW spectrum. The Bickley jet employed in the present study provides a more realistic representation of the EUC than the shear layer, and allows


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

us to extend the results of Sutherland by looking explicitly at the reflection, absorption and transmission of energy at the lower critical level. In future publications, we will extend this model to include effects of the surface boundary and the SEC, as well as the seasonal thermocline. Here, we focus on the effects of asymmetric stratification in isolation from such complications. We approach the stability of the asymmetrically stratified jet as a problem of general relevance, as well as focussing on those aspects most germane to the EUC. Background flow profiles used in the analyses are described in section 2. Section 3 contains the main results of the investigation. Linear stability analyses are used to establish the spectrum of normal mode instabilities at various values of stratification parameters. In section 4, the results are tested in the nonlinear regime by means of two-dimensional numerical simulations. The results are compared with observations of gravity waves the EUC in section 5. Section 6 contains a summary and concluding discussion.

2.5 2 1.5 1 0.5

(a) J

(b)

(c)

U

z

0 -0.5 -1 -1.5 -2 -2.5 0 0.5 1 0 0.1

J

L

U

N

0.2 2

0.3

0

1

2

Ri

Figure 2: Background profiles defining the asymmetrically
stratified jet. (a) Velocity; (b) Buoyancy gradient; (c) gradient Richardson number. For this case, JL = 0.25 and JU = 0.05.

2

Background states

In a fluid where density is controlled only by temperature, is proportional to the temperature deviation. The initial nondimensional buoyancy field is defined (up to an arbitrary additive constant) by

J , z > 0 Our mathematical model employs the Boussinesq equa U (x, y , z , 0) = (3) N2 tions in a physical space measured by the Cartesian coJ , z 0. z L ordinates {x , y , z }. For most analyses, flow will be restricted to the x and z directions. Asterisks are used The constants JU and JL are bulk Richardson numbers to indicate dimensional quantities; all other quantities are that characterize upper and lower layer stratification in nondimensionalized using velocity scale uo and the length comparison with the shear associated with the jet. scale ho . The nondimensional velocity field U (x, y , z , t) is given at t = 0 by the standard Bickley jet:

3

Linear stability analyses of normal modes

U (x, y , z , 0) = sech z .

2

(1)

A nondimensional buoyancy, (x, z , t), is defined in terms of the density (x, z , t), its representative value o 3.1 Methods and the gravitational acceleration g as The stability analyses are based on the usual Taylor - o g ho =- (2) Goldstein equation for linear normal modes of an inviscid, â 2. o uo 4


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

stratified, parallel, Boussinesq shear flow: ^ d2 ^ + 2 (z ) = 0; dz 2

tions: ^ w = i ^ (4) ^ u = -z ^ ^ = (u - c)z - uz ^ ¯ ¯^ ^ = -(2 + 2 ) ^ (8)

2 (z ) =

N d U /dz - - 2 . (U - c)2 U -c

2

2

2

(5)

The streamfunction has been assumed to have the normal mode form: ^ (x, z , t) = (z ) + µ (z ) exp[i(x - ct)] (6)

u ¯ K =- ·uw +w - w. (9) t z z The overbar indicates a horizontal average and primes denote fluctuations about that average. K = 1 (u 2 + w 2 ) 2 is the perturbation kinetic energy and is a scaled pressure. The remaining quadratic quantities, u w , w and w correspond to vertical fluxes of horizontal momentum, buoyancy and kinetic energy. To compute the flux profiles, the real parts of the relevant eigenfunctions are To minimize boundary effects, we impose the radiation multiplied, then averaged over one wavelength in x using ^ ^ condition d /dz = i at both upper and lower bound- the standard relation a b = 1 Real(a^). ^b 2 aries, with the sign of chosen in each case to eliminate A useful quantity for the characterization of energy parincoming waves (Hazel, 1972; Sun et al., 1998). The re- titioning is the flux Richardson number, defined here as sulting eigenvalue problem is solved numerically using w dz Rf = - . (10) the multiple shooting method of Hazel (1972). Because u w dz linear theory does not specify absolute mode amplitudes, The momentum flux u w is of additional interest because, we normalize the eigenfunctions so that at second order in µ, it affects the mean flow via the evolution equation ^ max = 1. (7) z u ¯ =- uw. (11) t z Eigenfunctions for perturbation quantities other than the This effect is of central importance in the application of streamfunction are recovered using the polarization rela- our results to the EUC. 5

in which µ is an ordering parameter and and c are the nondimensional streamwise wavenumber and phase velocity. Only the real parts of the perturbations are physically relevant. We assume that the modes of primary interest are two-dimensional (i.e. the wavevector is parallel to the background flow), but oblique modes are obtained easily via Squire's transformation (Drazin and Reid, 1981) and will be discussed where relevant. U (z ) = -d/dz and N 2 = d/dz are the background profiles of velocity and buoyancy gradient (squared buoyancy frequency) as given by (1) and (3).

-N 2 ^ ^ = U -c For each set of background profiles (i.e. for each set of values for JL and JU ), we scan over to identify classes of unstable modes. We then choose representative members of each class for more detailed study. The typical choice is the mode for which the exponential growth rate, given by = ci where ci is the imaginary part of the phase speed, is a maximum. Energetics of the linear, inviscid model are described by the perturbation kinetic energy equation:


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

5

(a)

0.15

(a)

S



0.05

z

0.1

0

0

-5
0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

0.5 V

1

-10

0

10

-130

0

130

5

(b)
1

(b)
0.8 0.6

0

c

r

0.4 0.2 0

z

-5
0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

0.5

1

-10

0

10
3

-70

0

70
3

a, p/2

Energy Flux â 10

Momentum Flux â 10



Figure 4: Eigenmodes of the most unstable sinuous (a) and Figure 3: Nondimensional growth rate (a) and phase speed
(b) as a function of nondimensional wavenumber for unstable modes of the classical Bickley jet (JL = JU = 0). Symbol size varies linearly with phase speed to aid in identification. Bullets indicate the fast growing mode in each class. varicose (b) modes for the unstratified Bickley jet (JL = J
U

=

0). Leftmost column gives the magnitude of the eigenfunction ^ (z ) (solid curve), normalized by its maximum value, as well as the phase normalized by 2 (dashed curve). Remaining columns show vertical fluxes of energy p w and momentum u w , respectively. Shaded boxes indicate the region between the upper and lower critical levels.

3.2 Results
As a point of departure, we briefly review the sinuous and varicose instabilities of the Bickley jet with JL = JU = 0 (Drazin and Reid, 1981). Figure 3 shows spectra of the growth rate and phase speed, indicating the existence of the two classes of unstable modes whose properties vary continuously as a function of wavenumber. Each class exhibits a well-defined wavenumber at which the growth rate is a maximum (bullets on figure 3). The sinuous mode dominates at all wavenumbers, and attains its maximum growth rate near = 1 (figure 3a). The varicose mode is most unstable at twice the wavelength, and exhibits higher phase speeds than the sinuous mode. Figure 4 shows the eigenfunctions of the fastest-growing members of each mode class. The streamfunction of the sinuous mode (figure 4a) is even with respect to z = 0, whereas the varicose mode is odd, i.e. undergoes a phase shift at z = 0. Each mode extracts energy from the mean flow near both the 6 upper and lower critical levels, as is evident from the energy flux profiles in figure 4. Each mode is configured so as to flux momentum outward away from the jet core, thus tending to broaden and weaken the jet. We now begin our discussion of stratification effects with a survey of the growth rate of the fastest growing mode at each point on the (JL , JU ) plane (figure 5). The special case JU = JL was investigated by Hazel (1972) and, as a limiting case, by Sutherland and Peltier (1992). As JU = JL increases, the growth rate drops monotonically and vanishes near JU = JL = 0.13. The nature of the dependence of the growth rate on stratification changes when either JL or JU (but not both) exceeds a critical value, which ranges between 0.17 and 0.22 (thin curves on figure 5). In this regime, the growth rate becomes nearly independent of the larger of the two strati-


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

(JL,JU)
0.4

(a)

0.1 0.08

S

U

J



0.35

=

L

J

0.06 0.04

0.3

0.02 0 0 0.2

R
0.4 0.6

V
0.8 1 1.2 1.4

0.25
1

JU

(b)
0.2
0.8 0.6

0.15 0.06 0.1 0.12 0.05 0.15 0 0 0.05 0.1 0.09 0.15 0.2 0.25 0.3 0.35 0.4 0.03 0

c
0.4 0.2 0 0

r

0.2

0.4

0.6



0.8

1

1.2

1.4

Figure 6: Nondimensional growth rate (a) and phase speed
(b) as a function of nondimensional wavenumber for unstable modes of the asymmetrically stratified jet with J J
U L

JL

= 0.09 and

= 0. Labels indicate the mode classes. Symbol size varies

Figure 5: Growth rate of the most unstable mode as a func- linearly with phase speed.
tion of J
L

and JU . Thin curves indicate the boundaries be-

yond which instability becomes approximately independent of the larger of JL and JU . Bullets indicate points at which more detailed analyses are described later in this section.

fication parameters. Instability persists at arbitrarily large values of the stratification in either the upper or the lower half-space, as long as the stratification in the other half space is sufficiently weak. It is this asymmetric stratification regime that is of primary interest to us in this paThe reason for the preferential stabilization of the V per. We will demonstrate that instability in this regime is mode is evident upon inspection of the energy flux profiles driven by a previously undiscovered mechanism. in figures 7a and b (second column). Not surprisingly, We first examine the effect of adding weak stratifica- both the S and V modes have lost their symmetry about tion to the lower half-space by setting JL equal to 0.09. z = 0. The S mode now extracts energy from the mean The analysis now reveals three distinct classes of unstable flow preferentially at the upper critical level, while the V modes (figure 6). Eigenfunctions for the fastest-growing mode extracts energy mainly at the lower critical level. member of each class are shown in figure 7. Two of the The V mode is thus affected more strongly by the stable mode classes are extensions of the sinuous and varicose stratification in the lower half-space. This distinction is modes of the unstratified case, and we will refer to them also evident in the buoyancy flux profiles (fourth column), as the S and V modes, respectively. The presence of strat- which indicate that the V mode does more work against 7

ification acts to damp the S mode slightly and the V mode more strongly (figure 6). In addition, growth rate maxima for both modes are shifted to higher wavenumbers. Both modes disappear at low wavenumbers and are replaced by a new mode class, denoted "R". The existence of this new mode is the main result of the present investigation, and its characteristics will be described presently.


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

5

(a)

S

substituting (11), and assuming that the flux vanishes at the boundaries, we obtain d = dt u w dz . udz ¯ (13)

0

z

-5 5

0

0.5 V

1

-40

0

40

-180

0

180

-40 -30 -20 -10

0

(b)

0

-5 5

0

0.5 R

1

-8

0

8

-170

0

170

-120

-80

-40

0

(c)

0

-5

0
a

0.5

1

-2

0

2
3

-20 -10

0

10 20
3

-3

-2

-1

0
3

The integral of the momentum flux thus indicates the direction in which the centroid moves as the mode grows to finite amplitude. The flux due to the S mode tends to raise the centroid of the jet. This finite-amplitude effect moves the shear away from the stratified region, the overall effect of which is to lower the Richardson number and enhance the growth of the instability. In contrast, the V mode acts to move the jet downward into the stratified region. The new mode class seen in figure 6 resembles the S mode in that it extracts energy from the mean flow primarily near its upper critical level (figure 7c). As a result, it does relatively little work against gravity. The new mode differs most dramatically from both the S and V modes in the region below the lower critical level: it drives strong fluxes of energy, momentum and buoyancy downward into the lower half plane. The tilt in the phase profile of the eigenfunction indicates a uniform phase progression with depth, indicative of wavelike signal propagation. In other words, the phase speed and wavelength of this mode are such that it excites an IGW field in the lower half plane having a real vertical wavenumber. For this reason, we refer to the new instability as the R (for "radiating") mode.

z

z

, /2
p

Energy Flux â 10

Momentum Flux â 10

Buoyancy Flux â 10

Figure 7: Eigenmodes of the most unstable S (a), V (b) and R
(c) modes for the asymmetrically stratified jet with J
L

= 0.09

and JU = 0. Leftmost column gives the magnitude of the eigen^ function (z ) (solid curve), normalized by its maximum value, as well as the phase normalized by 2 (dashed curve). Remaining columns show vertical fluxes of energy p w , momentum u w and buoyancy w , respectively. Shaded boxes indicate the region between the upper and lower critical levels.

Note that the amplitude of the R mode is not uniform in the region below the jet; rather, it decays exponentially with depth. This behavior should not be confused with evanescence, the decay of signals that do not resonate with any propagating mode. The decay seen here is due to the fact that the source of the radiation is a disturbance that grows exponentially with time. The wave amplitude also z udz ¯ = , (12) grows exponentially with time at any fixed depth. At a udz ¯ fixed time, however, the signal is strongest at points nearwhere the integrals cover the (possibly infinite) vertical est to the source. Exponential growth is also responsible extent of the flow. Differentiating with respect to time, for the nonzero buoyancy flux carried into the lower half 8

gravity, relative to its amplitude, than does the S mode. In every case, the buoyancy flux is strongly focussed at the lower critical level. The flux Richardson numbers of the V and S modes are 0.44 and 0.16, respectively. Note that the momentum flux due to the V mode (figure 7b, third column) is nearly flat except near the critical levels. This is a general characteristic of weak instabilities (Eliassen and Palm, 1961). The asymmetry of the momentum flux has an interesting implication for the motion of the jet as a whole. The centroid of the velocity distribution is


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

0.1

5

(a)
0.08 0.06

(a)

R

0



0.04 0.02 0

R

S
(b)

z

-5 5

0

0.5 S

1

-20 -10

0

10 20

-60

0

60

-12

-8

-4

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0

1

(c)
0.8

z

-5
0.6

0

0.5

1

-120

0

120

-280

0

280

-60 -40 -20

0

cr

a, p/2

Energy Flux â 103

Momentum Flux â 103 Buoyancy Flux â 103

0.4 0.2 0

Figure 9: Eigenmodes of the most unstable R (a) and S (b)
0 0.2 0.4 0.6



0.8

1

1.2

1.4

modes for the asymmetrically stratified jet with JL = 0.25 and = 0. Leftmost column gives the magnitude of the eigen^ function (z ), normalized by its maximum value, as well as
U

J

Figure 8: Nondimensional growth rate (a) and phase speed
(b) as a function of nondimensional wavenumber for unstable modes of the asymmetrically stratified jet with J J
U L

the phase normalized by 2 . Remaining columns show vertical fluxes of energy p w , momentum u w and buoyancy w , respectively. Shaded boxes indicate the region between the upper and lower critical levels.

= 0.25 and

= 0. Labels indicate the mode classes. Symbol size varies

linearly with phase speed.

increased to 0.28, indicating that the mode expends a significant fraction of the energy it gains from the mean flow plane by the R mode. Next, we examine the effect of stronger stratification in in doing work against gravity. The modes shown in figure 9 clearly illustrate our the lower half-space, the case JL = 0.25. The V mode is now entirely stabilized, and the growth rate of the fastest- hypothesized mechanism in which gravity waves are growing R mode is faster than that of the fastest-growing launched by instability on the upper flank of the jet, propS mode (figure 8). Turning to the energy flux profiles (fig- agate downward through the jet core, and are absorbed at ure 9, second column), we see that the R mode still ra- the lower critical level. Note in particular the intense didiates energy into the lower half plane as in the previous vergence of the momentum flux just above the lower criticase, but that energy flux is much weaker than before (rel- cal level of the S mode (figure 9b, third column). At finite ative to the amplitude of the perturbation streamfunction, amplitude, this divergence will lead to a rapid deceleracf. figure 7c). Although the wavelength and phase speed tion of the mean flow and consequent upward migration allow the R mode to resonate with a free IGW mode, very of the critical level in analogy with orographic waves in little of the mode's energy is able to penetrate beyond the the atmosphere (Fritts, 1989). This mechanism may aclower critical level. The S mode also fluxes very little en- count for turbulence observed on the lower flank of the ergy into the lower half-space and, as before, it does not EUC (figure 1d) and may play an important role in the resonate with any vertically propagating mode. For both zonal momentum balance. modes, the buoyancy flux is focussed near the lower critiHaving looked closely at the specific cases JL = 0, cal level. The flux Richardson number of the S mode has JL = 0.09 and JL = 0.25, we now take a broader 9


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

0.2

(a)
0.15

S

0.1 0.05 0

V

R

0.8

(b)
0.7

to be destabilized as soon as JL is increased above zero (though it is difficult to detect numerically at very small JL ). The R mode becomes the dominant instability for JL > 0.22. The growth rate continues to increase with increasing stratification, up to and beyond the largest JL value shown here. The reasons for this relative freedom from buoyancy damping will be explored below. Both the R and V modes have much larger phase speeds than the S mode, indicating that their critical layers are located close to the jet core, while the critical layers of the S mode lie further out on the flanks of the jet (figure 10b).

(c)
1

For all modes, the dominant wavenumber increases slightly with increasing stratification. The R mode is the 0 longest, exceeding the S mode in wavelength by a factor 1.5 of three or more at all JL (figure 10c). This difference (d) 1 accounts for the R mode's ability to couple efficiently with freely propagating gravity waves in the lower half0.5 space. In fact, the R mode is the only mode that cou0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 ples to a free mode with real vertical wavenumber (figure JL 10d). The vertical wavenumber at which the R mode radiFigure 10: Scalar properties of the fastest growing modes as ates increases strongly with increasing stratification. Over functions of stratification parameter JL for the case JU = 0. the range shown in figure 10, the vertical wavenumber is Thin, thick and dashed curves indicate the fastest-growing S, R about twice the horizontal wavenumber.
0.5

and V modes, respectively. (a) Growth rate. (b) Phase speed. (c) Streamwise wavenumber. (d) Real vertical wavenumber of the

The local Richardson number at the lower critical level, RiL , varies approximately in proportion to JL , and is corresponding free mode, given by (5). nearly the same for all three modes. Because the V mode grows by extracting energy from the mean flow at the lower critical level, we expect that it will be damped for view of the dependence of the stability characteristics of all RiL > 1/4. In fact, RiL =0.18 at the largest JL for the asymmetrically stratified jet upon the lower stratifica- which the V mode is unstable. For larger JL , only the two tion parameter JL . Figures 10, 11 and 12 show various modes that extract energy at the upper critical level are scalar properties of the S, V and R mode classes as func- unstable. In general, instability in strongly stratified cases retions of JL with JU = 0. For each class, the mode illustrated is that which maximizes the growth rate. The quires that modes be configured so as to expend only a S and V modes are both stabilized by the stratification: small fraction of their energy uptake in doing work against the former is damped entirely for JL > 0.3, while the gravity. Because its energy source is in the stratified relatter is damped for JL > 0.1. The R mode appears gion, the V mode expends a large fraction of its energy 10

r



cr

0.6 0.5 0.4

1.5


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

(a)

1

(a)

100

RiL


(b)

0.5

50

0

0 0.06 0.04

(b)

1

Rf

0.5


0.02 0 0

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

J

L

J

L

Figure 11: Scalar properties of the fastest growing modes as
functions of JL for the case J
U

Figure 12: Scalar properties of the fastest growing R mode
as functions of JL for the case J
U

= 0. Thick, thin and dashed

= 0. (a) Angle between

curves indicate the fastest-growing S, R and V modes, respectively. (a) Gradient Richardson number at the lower critical level. (b) Flux Richardson number.

the wavevector and the x axis. (b) Growth rates of the fastest growing mode (dashed) and the fastest growing two-dimensional mode (solid).

working against gravity. By the time the mode disappears at JL = 0.1, its flux Richardson number is about 0.5 (figure 11b). The flux Richardson number of the S mode increases more slowly, reaching a value 0.38 at the largest JL for which the instability exists. In contrast, the R mode is configured so as to do relatively little work against gravity even at the largest JL values included here, and is therefore able to grow despite strong stratification in the lower half plane.

wavevector. Oblique modes are therefore expected to occur in pairs, having equal growth rates and phase speeds but directed at opposite angles from the streamwise direction. Smyth et al. (1988) noted that the fastest growing mode is oblique in any region of parameter space where the growth rate increases faster than J 1/2 (where J is a bulk Richardson number). In the present case, the growth rate of the R mode in1/2 creases faster than JL for all JL less than 0.25. Figure 12a shows the angle by which the wave vector of the fastest growing mode departs from the x axis. This angle is large for small JL , but decreases to zero by JL = 0.25. The R mode is therefore two-dimensional over most of the large JL values for which it is the fastest growing instability. An exception to this occurs in the small range 0.22 < JL < 0.25, where a three-dimensional R mode dominates. However, the difference in growth rate between this and the corresponding two-dimensional mode is less than one half percent (figure 12b). In the nonlinear simulations described in the next section, we will examine two-dimensional modes only.

The fact that the R mode is destabilized by stratification raises the possibility that the fastest growing mode may be oblique, i.e. it may have a nonzero spanwise component to its wavevector. In unstratified flow, Squire's theorem (Drazin and Reid, 1981) shows that, while they always exist, no oblique mode can be the fastest growing. In the presence of stratification, however, Squire's theorem is modified so that the growth rate of a threedimensional mode is related to that of a two-dimensional mode at higher bulk Richardson number (e.g. Smyth and Peltier, 1990). Any mode class whose growth rate increases with increasing stratification has the potential to have an oblique mode as its fastest growing member. As We complete the description of the linear normal modes long as the mean flow is parallel, the growth rate is al- of the asymmetrically stratified jet by returning briefly to ways even with respect to the spanwise component of the the case in which the upper level stratification parame11


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

(a)

0.2

5 R



0.1

0

0

(b)

1 0.8 0.6 0.4

z

cr

-5

0

, /2
a p

0.5

1

-7

0

7
3

-60

0

60

-30
3

-20

-10
3

Energy Flux â 10

Momentum Flux â 10 Buoyancy Flux â 10

(c)

1.5 1 0.5 0

Figure 14: Eigenmodes of the most unstable R mode for the
asymmetrically stratified jet with J = 0.05. ^ Leftmost frame gives the magnitude of the eigenfunction (z ),
L U



= 0.25 and J

(d)

1

normalized by its maximum value, as well as the phase normalized by 2 . Remaining frames show vertical fluxes of energy p w , momentum u w and buoyancy w , respectively. Shaded boxes indicate the region between the upper and lower critical levels.

r
(e)

0.5

0 1

Rf RiU RiL
(f) (g)

0.5

0 0.12

0.1

0.08 1

0.5

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

J

L

Figure 13: Scalar properties of the fastest growing modes as
functions of J
L

for the case J

U

= 0.05. Thick, thin and

dashed curves indicate the fastest-growing S, R and V modes, respectively. (a) Growth rate. (b) Phase speed. (c) Streamwise wavenumber. (d) Real vertical wavenumber of the corresponding free mode. (e) Flux Richardson number. Gradient Richardson number at the upper (f) and lower (g) critical level.

ter JU is nonzero. Scalar parameters for JU = 0.05 are shown as functions of JL in figure 13. Comparing with the case JU = 0 (figures 10 and 11), we see that the growth rates are reduced by the upper level stratification. The S and V modes are damped at lower values of JL , and the asymptotic growth rate of the R mode at large JL is reduced by about one half. The phase speeds of all modes are increased somewhat (figure 13b), indi-

cating that their critical levels have moved toward the jet core. The streamwise wavenumbers at which the modes are most unstable are not significantly changed from the JU = 0 case (figure 13c), but the vertical wavenumber at which the R mode resonates is reduced (figure 13d). Because the stratification is now nonzero at the upper critical level, the two modes (S and R) that extract their energy from the mean flow in that region do more work against gravity, with the result that their flux Richardson numbers are generally higher (figure 13e). The gradient Richardson number at the upper critical level is now nonzero in all cases, but it remains substantially smaller than 1/4 for each unstable mode (figure 13f). The gradient Richardson number at the lower critical level (figure 13g) is slightly reduced, but it retains the property of being essentially equal for the fastest growing mode of each class. The R mode at JL = 0.25, JU = 0.05 now shows a significant buoyancy flux at the upper critical level, but its tendency to radiate energy into the lower half-space is unchanged from the JU = 0 case (figure 14, cf. figure 9a).

12


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

4

Nonlinear simulations

The unstable normal modes described in the previous section have the potential to grow to large amplitude, and thus to generate large fluxes of energy and momentum and a vigorous turbulent energy cascade. Since linear theory tells us nothing about the amplitude a given instability will attain, it does not allow us to evaluate the strength of these processes in any real geophysical system. To estimate the actual fluxes and mixing rates requires solution of the fully nonlinear initial value problem. To do this in detail, particularly for a flow of geophysical scale, is beyond the scope of the present study. However, a strongly constrained nonlinear model offers the opportunity to make useful first estimates.

= - t z x x z 1 2 ¯ + ( - ) + D (x, z , t) ReP r =
2

(15)



(16)

The fields (x, z , t) and (x, z , t) represent vorticity and the streamfunction, respectively, while (x, z , t) represents the buoyancy as before. The Reynolds number Re = uo ho / represents the reciprocal of the nondimensional kinematic viscosity. P r denotes the Prandtl number, the ratio of viscosity to scalar diffusivity. For the present experiments, we choose Re = 300 and P r = 1. Note that the diffusion operators are not applied to the horizontally-averaged flow (denoted by overbars). D represents a radiative damping term to be described beBecause the dominant linear instabilities of parallel low. shear flows tend to be two-dimensional, it makes sense The numerical model approximates solutions to the inito investigate their evolution to large amplitude via simu- tial value problem defined by (14-16), together with perilations in which motions in the third dimension are sup- odic boundary conditions in x and flux-free boundary conpressed. In these simulations, the effects of small-scale ditions at the upper and lower limits of the computational turbulence are represented only crudely via a diffusion op- domain. Spatial discretization is Fourier pseudospectral erator. Aspects of the flow for which these motions are in the horizontal, second-order centered difference in the crucial, such as turbulent mixing at critical levels, cannot vertical. Temporal discretization is via a second-order be evaluated with any confidence. However, these simu- Adams-Bashforth method. lations are expected to deliver realistic amplitudes for the In order to mimic the radiation boundary conditions large-scale aspects of the flow evolution, in particular the employed in the linear analyses (section 3.1), the compufluxes due to wave propagation. tational domain is divided into an inner "physical" layer and outer sponge layers. In the sponge layers, Rayleigh damping is applied to the scalar field so as to minimize the 4.1 Methods reflection of gravity waves back into the physical layer. The damping coefficient increases smoothly from zero at To take full advantage of the restriction to two dimenthe edge of the physical layer, reaching a maximum value sions, the nondimensional Boussinesq equations are cast of = 0.20 at the boundary of the computational domain, in vorticity-streamfunction form as: e.g.: = - t z x x z 12 + ( - ) ¯ + x Re D = - (14) zb - z 1 ¯ 1 - cos ( - ) 2 zb - zB (17)

where zb and zB denote the lower boundaries of the physical and computational domains, respectively. Within the 13


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

physical domain, D = 0. The vertical grid spacing is uniform within the physical domain, but varies in the sponge layers. The grid interval increases linearly from its physical-domain value at the inner edge of the sponge layer to three times that value at the domain boundary. Flow diagnostics are computed over the physical domain only.

0.025 0.02 0.015 K 0.01


Initial conditions consist of basic states as defined in 0.005 section 2, with added perturbations proportional to the unstable normal modes computed in section 3. The perturba0 0 50 100 150 200 tion amplitudes are chosen so that the perturbation kinetic t -4 energy is 10 times the kinetic energy of the mean flow at t = 0. The physical domain is defined by 0 x < L; Figure 15: Volume-averaged perturbation kinetic energy as a -5 z 5, where L is some integer multiple of the function of time for simulations S14 (thick solid), R25 (thin wavelength of the eigenfunction. Upper and lower sponge solid) and R2505 (dashed). layers extend to z = 9 and z = -13. We have used this methodology to investigate the nonlinear evolution of many of the unstable modes described in the previous section. Here, we describe three particular cases that encompass the range of nonlinear behaviors found. In each case, the initial condition consists of a mean flow specified by the values of JL and JU , plus a small perturbation proportional to the eigenfunction of some linear normal mode. In the first case, the mean flow is characterized by JL = 0.14, JU = 0, so that stratification in the lower half-space is significant but the S mode still dominates. This run is designated S14. The second case, denoted R25, has JL = 0.25, JU = 0, so that the R mode dominates. For the third simulation, we retain JL = 0.25 but increase JU to 0.05, to see the effect of stratification in the upper half space. This run is denoted R2505.

increase in kinetic energy. For case S14, K grows to a maximum near t=50, then decays slowly. The radiating mode with JL = 0.25 and JU = 0 (R25) grows more slowly at first, but exhibits a sharp increase in growth rate after about t=60 and eventually attains larger energy than the S case. The reason for this late growth phase will be explored below. The R mode with JL = 0.25 and JU = 0.05 (R2505) grows slowly throughout the simulation. The temperature, vorticity and perturbation streamfunction fields for the S mode at t=40 and t=80 are shown in figure 16. Three wavelengths of the primary instability are included in the computational domain to facilitate subsequent comparison with the longer R modes. The counterclockwise (positive) vorticity of the upper flank of the jet is concentrated into discrete monopoles, while the lower flank vorticity appears as curved braids (figure 16, middle frames). The maximum clockwise vorticity (thick contours) is larger than the maximum counterclockwise vorticity (thin contours), due to the baroclinic torque exerted by the buoyancy force. The upper edge of the stratified fluid is distorted into smooth ripples (figure 16, left

4.2 Results
The evolution of the perturbation kinetic energy, spatially averaged over the physical domain, is shown for our three cases in figure 15. Not surprisingly, the modes with the largest linear growth rates exhibit the most rapid initial

14


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum


5 (a)
min, max, interval = -1.800 0.008 0.069


min, max, interval = -1.180 0.761 0.197


min, max, interval = -0.214 0.238 0.011

t=40

0

z

-5 5 (b)
min, max, interval = -1.800 0.008 0.069 min, max, interval = -0.965 0.744 0.161 min, max, interval = -0.268 0.300 0.014

t=80

0

z

-5

0

5 x

10

15

0

5 x

10

15

0

5 x

10

15

Figure 16: Scaled temperature, vorticity and perturbation streamfunction fields for run S14 at t=40 (a) and t=80 (b). The perturbation streamfunction is defined relative to the horizontally-averaged profile. Shading indicates the original locations of the critical levels. For the vorticity and the streamfunction, thick (thin) contours indicate negative (positive) values.

hand frames). Further down, however, the scalar field exhibits large amplitude overturns. This enhanced displacement begins at the lower critical level, and indicates intense wave-mean flow interaction in that region (figure 16a, left frames, bottom of shaded region). Enhanced turbulence observed on the lower flank of the EUC (figure 1d) may result from a similar mechanism. By t=80, (figure 16b), the overturning region has migrated upward due to wave-induced acceleration of the mean flow. Beyond this point in the flow evolution, it is likely that a fully three-dimensional simulation would reveal motions in the spanwise direction. Secondary stability analyses of shear-driven billows (e.g. Klaassen and Peltier, 1985b; Sutherland et al., 1994) have revealed that threedimensional motions become important once K reaches

its maximum value. Analyses of waves approaching a critical level by Winters and D'Asaro (1994) have shown a similar result: overturns break down partly via threedimensional convective motions. Ultimately, our twodimensional flow approaches a state of nonlinear equilibrium, with continued slow evolution due only to diffusion. The R mode (figure 17) is distinguished from the S mode by its large wavelength (cf. figure 10c) and slow growth (figure 10a, figure 15). At t = 40, flow distortions due to the R mode are relatively weak. The perturbation streamfunction exhibits wave radiation into the lower half plane, as expected on the basis of the stability analyses. By t = 60, the disturbance has reached large amplitude and exhibits strong vorticity and distinct overturning near the lower critical level. The long wave is now modulated

15


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum


min, max, interval = -0.000 3.220 0.124


min, max, interval = -1.230 0.840 0.205


min, max, interval = -0.212 0.188 0.010

(a) 4 2 0 -2 -4 z

t=40

min, max, interval = -0.000 3.230 0.124

min, max, interval = -1.430 0.829 0.238

min, max, interval = -0.328 0.304 0.016

(b) 4 2 0 -2 -4 z

t=60

min, max, interval = -0.000 3.230 0.124

min, max, interval = -1.810 0.841 0.302

min, max, interval = -0.860 0.540 0.041

(c) 4 2 0 -2 -4 0 z

t=80

5 x

10

15

0

5 x

10

15

0

5 x

10

15

Figure 17: Scaled temperature, vorticity and perturbation streamfunction fields for run R25 at t=40 (a), t=60 (b) and t=80 (c). The
perturbation streamfunction is defined relative to the horizontally- averaged profile. Shading indicates the original locations of the critical levels. For the vorticity and the streamfunction, thick (thin) contours indicate negative (positive) values.

by a disturbulence of shorter wavelength. By t = 80, the character of the disturbance that originated with the R mode has changed dramatically. The flow is now dominated by a large amplitude, short wavelength disturbance. Strong overturning is evident. The short-wave disturbance has caused long plumes of stratified fluid to be ejected into the upper half plane. Not long after this, the jet is completely disrupted by the large am-

plitude disturbance. (By this point, the flow is certain to be influenced by three-dimensional effects, so the present results must be treated as preliminary. It seems unlikely, however, that three-dimensional motions would alter the main features of the flow.) Downgoing radiation in the lower half-space is still present at t = 80, though it is weak in comparison with the intense vortices above the jet and is therefore not visible in figure 17c.

16


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum


min, max, interval = -0.000 3.700 0.142


min, max, interval = -1.120 0.891 0.187


min, max, interval = -0.132 0.124 0.006

(a) 4 2 0 -2 -4 z

t=80

min, max, interval = -0.000 3.700 0.142

min, max, interval = -1.290 0.943 0.215

min, max, interval = -0.258 0.228 0.012

(b) 4 2 0 -2 -4 0 z

t=160

5 x

10

0

5 x

10

0

5 x

10

Figure 18: Scaled temperature, vorticity and perturbation streamfunction fields for run R2505 at t=60 (a) and t=120 (b). The
perturbation streamfunction is defined relative to the horizontally- averaged profile. Shading indicates the original locations of the critical levels. For the vorticity and the streamfunction, thick (thin) contours indicate negative (positive) values.

R mode with R25 is much slower at first, but increases rapidly after about t=60. This displacement tends to reduce the effect of the stratification. On the basis of our earlier linear stability analyses (e.g. figure 10a), we expect that this will destabilize shorter modes similar to the S mode. (This is confirmed by separate linear stability analyses of the case where the discontinuity in stratification is located below the center of the jet.) This secondary effect of jet migration is the reason for the appearance of The vertical motion of the jet provides a clear indica- the short-wave instability in the R25 case (figure 17) and tion of wave-mean flow interactions. The location of the for the complex evolution of the perturbation kinetic envelocity centroid as a function of time is shown for all ergy (figure 15). three simulations in figure 19. The S mode at JL = 0.14 displaces the jet upward, as expected on the basis of the Although the catastrophic breakdown of the jet coinmomentum flux of the linear eigenmode (cf. figure 7 and cides with the growth of the short-wave secondary inaccompanying discussion). Displacement of the jet by the stability, the long-wave disturbance is also essential to With the addition of nonzero stratification in the upper half-space (case R2505), the growth of the R mode is slowed considerably. Not until t 80 does significant overturning develop at the lower critical level (figure 18). The secondary instability and subsequent catastrophic breakdown observed in the R25 case is prevented. As in the case R25, wave radiation into the lower halfplane is evident. 17


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

2
momentum flux

0.025 0.02 0.015 0.01 0.005 0

(a)

1.8 1.6 1.4 1.2 1

-0.005 x 10
-4

0.8 0.6
momentum flux 1 0.5 0 -0.5 -1 -1.5 -2 0

(b)

0.4 0.2 0 0 50 100 t 150 200

Figure 19: The centroid of the mean velocity distribution as a
function of time, showing the vertical migration of the jet center due to the net momentum flux of the unstable mode. Curves correspond to cases S14 (thick solid), S22 (thin solid) and R2505 (dashed).

50

100 t

150

200

Figure 20: Vertical flux of horizontal velocity across (a) the
upper critical level and (b) z=-5. Curves correspond to cases S14 (thick solid), R25 (thin solid) and R2505 (dashed).

the process. A counterexample is provided by simulation S14, which was initialized with the short-wave mode alone. In this case, the disturbance equilibrates at finite amplitude without destroying the jet. In a separate experiment (not shown), the S14 simulation was seeded with a random long-wave disturbance in addition to the shortwave eigenmode. The long-wave component eventually grew to large amplitude and the jet was disrupted as in the R25 case. In run R2505, secondary instability was strongly suppressed by the nonzero stratification in the upper half space. We conclude this section with a look at the momentum fluxes that arise at two particular levels in our simulations. Fluxes across the upper critical level (figure 20a) correspond to strong instability and overturning above the jet, while fluxes across the lower boundary of the physical domain, z = -5 (figure 20b) represent wave propagation into the deep ocean. In the disturbance that develops from the S mode (S14), the momentum flux at the upper critical

level grows to a maximum at about the same time that the perturbation kinetic energy peaks (cf. figure 15, t = 50). Beyond this point, the disturbance settles into a nonlinear, quasi-equilibrium state in which the momentum flux across the upper critical level drops to zero. This vanishing of the momentum flux is expected, for otherwise the background jet would continue to migrate upwards (rather than remaining steady at 1 as shown in figure 19) and the nonlinear equilibrium state could not be sustained. In case R25, the momentum flux at the upper critical level increases to 0.01 shortly before the secondary instability sets in and the jet begins to migrate (cf. figure 19). The flux then exhibits large amplitude oscillations. When weak stratification is added to the upper half-space (case R2505), the momentum flux at the upper critical level is reduced by more than an order of magnitude. In the R25 and R2505 cases, there is a consistent downward momentum flux at z = -5 that grows along with the disturbance

18


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

amplitude. In contrast, the momentum flux at z=-5 for the 10c) or less. This corresponds to an approximate lower S14 case fluctuates about zero. bound on the zonal wavelength of 500m. For comparison, the observations of Moum et al. (1992) and Lien et al. (1996) and the analyses of Sun et al. (1998) all deliv5 Comparison with EUC observa- ered wavelengths in the range 200-350m. In contrast, the wavelength of the S mode is consistent with the observations tions. Zonal phase speeds of the modes computed by Sun The asymmetrically stratified jet represents an incomplete et al. tended to be near zero or even negative, while Lien et model of the EUC. It accounts for instability due to the al. estimated a phase speed of 0.2m/s. This is in contrast shear on the upper and lower flanks of the jet, and for with the phase speed of the R mode, which is generally its modification by the asymmetric stratification profile, greater than 0.6, or 0.6m/s in dimensional terms. but other important factors such as the surface current are The momentum fluxes shown in figure 20a may be omitted from the model. We now compare the charactercompared with the estimates made by Lien et al. (1996) istics of our computed modes with observational analyses, from observations of a large-amplitude wave signal above in order to assess the ability of this limited model to capthe core of the EUC. The maximum zonal stress associture the physics of the EUC. We compare our results with ated with that disturbance was estimated as 0.3N m-2 . the observational results of Moum et al. (1992) and Lien In our nondimensionalization, this becomes u w = 3 â et al. (1996), who observed large-amplitude waves on the 10-4 . The stresses our model develops above the jet exupper flank of the EUC, and also with stability analyses ceed this value by up to two orders of magnitude (figure of measured profiles from the EUC (Sun et al., 1998). 20). The most realistic model result is from run R2505, Appropriate length and velocity scales for the EUC are for which the momentum flux at the upper critical level ho = 40m and uo = 1m/s (figure 1). The squared approached 10-3 at the end of the simulation. Comparbuoyancy above the EUC is rarely greater than 10-4 s-2 , ison with the observations of Moum et al. (1992) yields whereas typical values just below the EUC vary between an even wider discrepancy, as their estimates of the mo10-4 and 10-3 s-2 . In our nondimensionalization, these mentum flux were an order of magnitude smaller still than levels of stratification become 0.16 and 1.6, respectively. those of Lien et al. In addition, accelerations due to the The domain of interest for the EUC is therefore approxmomentum flux convergence in the model are several orimated by 0 JU 0.16; 0.16 JL 1.6. In the ders of magnitude stronger than those driven by the zonal JL - JU plane shown on figure 5, this domain occupies a pressure gradient in the equatorial Pacific. rectangle in the lower right-hand corner that extends to the right beyond the largest value of JL shown. The R mode It therefore appears that our model develops momendominates over most of this stratification regime. For JU tum fluxes that are much more intense than those found greater than about 0.1, the flow is stable. on the EUC. The addition of stratification above the jet The maximum growth rate of the R mode occurs when JU vanishes, and is about 0.05. Scaled for the EUC, this gives an e-folding time of about thirteen minutes. This is comparable with the fastest growing modes found by Sun et al. (1998). Typical nondimensional values for the streamwise wavenumber of the R mode are 0.5 (figure damps the instabilities considerably, but even the case R2505 develops fluxes that would severely disrupt the EUC over a period of a few hours. This is in dramatic contrast with the EUC, whose large-scale structure varies only on timescales of months to years. These results indicate that instability governed by the main local structure

19


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

of the EUC is more than strong enough to account for the in the lower half plane, its lower critical level provides observed waves and turbulence; in fact, we must now ask a barrier against energy leakage at sufficiently high JL . why the EUC is not considerably more turbulent than it is. In the nonlinear regime, however, the present simulations reveal no significant change in the downward momentum flux as a function of stratification.

6

Summary

We have examined the normal mode instabilities of a Bickley jet in a fluid with asymmetric stratification. We have identified three classes of unstable modes: the S and V modes are extensions of the sinuous and varicose instabilities of the unstratified Bickley jet, while the third mode, denoted "R", has no counterpart in the unstratified case. Stable stratification in the lower half-space (i.e. JL > 0) damps both the S and V modes. For weak stratification (JL < 0.22), the S mode dominates. At larger JL , the fastest growing mode is the R mode, whose long wavelength enables it to couple with free IGW modes and thus to radiate energy downward away from the jet. This characteristic is strongly suggestive of the wave radiation observed at depth the in equatorial Pacific. For JL < 0.25, the R mode appears as a superposition of two oblique disturbances whose wave vectors point at equal and opposite angles from the x axis. The angle of the fastest growing R mode approaches 90 degrees at small JL , decreases as JL increases, and vanishes at JL = 0.25. Beyond this point, the dominant R mode is two-dimensional. For 0.22 < JL < 0.25, the fastest growing disturbance is a pair of R modes inclined at up to 20 degrees from the x axis. However, these modes grow only marginally faster than their two-dimensional counterparts. Therefore, while three-dimensional primary instability may exert some influence on flow evolution, it is unlikely to be dominant. The R mode is able to grow despite strong stratification because its energy is focussed on the upper flank of the jet, away from the stratification. Linear theory suggests that, although the R mode resonates with the IGW field

Nonlinear simulations show that both the S and R modes generate strong wave-mean flow interactions at the lower critical layer, leading to overturning of the temperature field that would trigger turbulence in a threedimensional flow. In both cases, momentum flux divergences cause the jet to migrate upwards away from the stratification. (This migration is reversed in the case of the V mode, but the latter is never dominant.) Migration is slower for the R mode than for the S mode, but when the jet does begin to migrate upward, the result is a dramatic change in the nature of the disturbance. The departure of the jet from the stratified zone destabilizes a short-wave instability similar to the S mode, which grows rapidly and greatly accelerates the breakdown of the jet. It appears that no amount of stratification in the lower half-space alone can prevent this breakdown, though weak stratification in the upper half space stabilizes the jet considerably. Both linear and nonlinear analyses support the hypothesis that strong turbulence and wave-mean flow interactions on the lower flank of the jet may result from the radiation of gravity waves from the upper flank. Waves are generated by the dynamic instability of the weakly stratified shear on the upper flank, propagate downward, and break upon encountering the critical level on the more strongly stratified lower flank. With respect to the EUC, the present results lead to an interesting conclusion: if the EUC were simply an isolated jet with stable stratification below and mixed fluid above, it would be torn apart by shear instability within a few hours. In reality, of course, the large-scale structure of the EUC varies only on timescales of months to years. We are thus left with the question: what stabilizes the EUC? A partial answer is provided by the addition

20


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

of stratification above the jet, but the momentum fluxes remain at least an order of magnitude larger than observations. In addition, the wavelength of the dominant R mode is longer by a factor of two than that of waves observed near the EUC, and its zonal phase speed is too large. Unrealistically large momentum fluxes can be reduced by increasing JU to values near the stability boundary, but the discrepancy in wavelength remains. We now evaluate the potential importance of several factors that are missing from the present model. · Three-dimensional motions: The main effect of three-dimensional turbulence is to smooth gradients via enhanced molecular mixing. In the present twodimensional simulations, the Reynolds number is set very low (of order 102 ). As a result, the effects of turbulent mixing are at least crudely represented and are, in fact, probably exaggerated.

· The South Equatorial Current: The presence of the counterflowing surface current increases the kinetic energy available to drive instability and may thus be expected to destabilize the flow. However, preliminary results indicate that the reverse is true; the surface current actually helps to stabilize the EUC. Also, the presence of the surface current should give rise to modes with small or negative zonal phase velocity, in better agreement with observations.

In summary, it seems likely that better correspondence with observations will result primarily from refinement of the background profiles and the upper boundary condition rather than from the inclusion of forcing or improved turbulence modeling. Future publications will describe extensions of the model to incorporate these factors, leading to improved understanding of waves and instability in the · Forcing: The EUC is sustained by a zonal pressure equatorial zonal current system. gradient. In our model, the forcing needed to sustain Acknowledgements: This work has benefited from disthe jet against instability would have to be several orcussions with Bruce Sutherland. The research was ders of magnitude stronger than that observed in the funded by the National Science Foundation under grant equatorial oceans. Also, the inclusion of such forcOCE9711872. In the published version of this paper, we ing would increase the momentum fluxes developed neglected to make reference to Skyllingstad and Denbo in the model, and those fluxes are already too large (1994). That work examined gravity waves and instabilto match the observations. ities of the EUC via large-eddy simulations, and was the · The seasonal thermocline: This thin layer of en- source of many of the ideas explored here. hanced stratification is located near or slightly below the EUC, and is expected to exert a stabilizing influence. · The ocean surface: Boundary proximity is known to stabilize long modes (e.g. Hazel 1972), and may therefore suppress the R mode that leads to the destruction of the jet in the present simulations. The wavelength of the R mode is much larger than the distance from the jet core to the surface, which means that the boundary proximity effect is likely to be significant. 21


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

References

Lindzen, R., 1981: Turbulence and stress due to gravity wave and tidal breakdown. J. Geophys. Res., 86, 9707.

Dillon, T., J. Moum, T. Chereskin, and D. Caldwell, 1989: Zonal momentum balance at the equator. J. Phys. Lindzen, R. and K. Tung, 1978: Wave overreflection and Oceanogr., 19, 561­570. shear instability. J. Atmos. Sci., 35, 1626­1632. Mack, A. and D. Hebert, 1997: Internal gravity waves in the upper eastern equatorial Pacific: Observations and numerical solutions. J. Geophys. Res., 102, 21081­ Eliassen, A. and E. Palm, 1961: On the transfer of energy 21100. in stationary mountain waves. Geophys. Publ., 22, 1­ 23. Moum, J., D. Caldwell, and C. Paulson, 1989: Mixing in the equatorial surface layer and thermocline. J. GeoFritts, D., 1989: A review of gravity wave saturation phys. Res., 94(C2), 2005­2021. processes, effects, and variability in the middle atmosphere. Pure Appl. Geophys., 130, 343­371. Hazel, P., 1972: Numerical studies of the stability of inviscid parallel shear flows. J. Fluid Mech., 51, 39­62. Moum, J., D. Hebert, C. Paulson, and D. Caldwell, 1992: Turbulence and internal waves at the equator. Part I: Statistics from towed thermistor chains and a microstructure profiler. J. Phys. Oceanogr., 22, 1330­ Hebert, D., J. Moum, C. Paulson, D. Caldwell, T. Chere1345. skin, and M. McPhaden, 1991: The role of the turbulent stress divergence in the equatorial pacific zonal Peters, H., M. Gregg, and J. Toole, 1989: Meridional varimomentum balance. J. Geophys. Res., 96, 7127­7136. ability of turbulence through the equatorial undercurDrazin, P. and W. Reid, 1981: Hydrodynamic Stability. Cambridge University Press, Cambridge.

rent. J. Geophys. Res., 94, 18003­18009. Klaassen, G. and W. Peltier, 1985b: The onset of turbulence in finite-amplitude Kelvin-Helmholtz billows. J. Qiao, L. and R. Weisberg, 1997: The zonal momentum Fluid Mech., 155, 1­35. balance of the equatorial undercurrent in the Central Pacific. J. Phys. Oceanogr., 27, 1094­1119. Lien, R.-C., D. Caldwell, M. Gregg, and J. Moum, 1995: Turbulence variability at the equator in the central paSkyllingstad, E. and D. Denbo, 1994: The role of internal cific at the beginning of the 1991-1993 El Nino. J. Geogravity waves in the equatorial current system. J. Phys. phys. Res., 100(C4), 6881­6898. Oceanogr., 24, 2903­2110. Lien, R.-C., M. McPhaden, and M. Gregg, 1996: Highfrequency internal waves and their possible relationship Smyth, W., G. Klaassen, and W. R. Peltier, 1988: Finite amplitude Holmboe waves. Geophys. Astrophys. Fluid to deep-cycle turbulence. J. Phys. Oceanogr., 26, 581­ Dyn., 43, 181­222. 600. Lin, C.-L., J. Ferziger, J. Koseff, and S. Monismith, 1993: Smyth, W. and W. Peltier, 1989: The transition between Simulation and stability of two-dimensional internal Kelvin-Helmholtz and Holmboe instability: an investigation of the overreflection hypothesis. J. Atmos. Sci., gravity waves in a stratified shear flow. Dyn. Atmos. 46(24), 3698­3720. Oceans, 19, 325­366. 22


Dyn. Atm. Oceans Submitted 17 July, 2001, accepted 8 May 2002.

W.D. Smyth & J.N. Moum

Smyth, W. and W. Peltier, 1990: Three-dimensional primary instabilities of a stratified, dissipative, parallel flow. Geophys. Astrophys. Fluid Dyn., 52, 249­261. Sun, C., W. Smyth, and J. Moum, 1998: Dynamic instability of stratified shear flow in the upper equatorial Pacific. J. Geophys. Res., 103, 10323­10337. Sutherland, B., 1996: Dynamic excitation of internal gravity waves in the equatorial oceans. J. Phys. Oceanogr., 26, 2398­2419. Sutherland, B., C. Caulfield, and W. Peltier, 1994: Internal gravity wave radiation and hydrodynamic instability. J. Atmos. Sci., 51, 3261­3280. Sutherland, B. and W. Peltier, 1992: The stability of stratified jets. Geophys. Astrophys. Fluid Dyn., 66, 101­131. Veronis, G., 1960: An approximate theoretical analysis of the equatorial undercurrent. Deep Sea Res., 6, 318­327. Winters, K. and E. A. D'Asaro, 1993: Two-dimensional instability of finite amplitude internal gravity waves near a critical level. J. Geophys. Res., 94, 12709­ 12719. Winters, K. and E. A. D'Asaro, 1994: Three-dimensional wave instability near a critical layer. J. Fluid Mech., 272, 255­284.

23