Internal-wave properties in weakly

stratified layers

by Theo Gerkema∗ and Eleftheria Exarchou†

Submitted to the Journal of Marine Research, July 1, 2008

ABSTRACTFollowing earlier studies, it is argued that internal waves propagating in weakly

stratified layers are affected by the Coriolis terms proportional to the cosine of lati-

tude (which are neglected in the so-called Traditional Approximation). A systematic

overview is given of the wave properties if these terms are included. The usage of

characteristics and vertical modes is compared and contrasted. It is shown that the

ratio of potential and kinetic energy varies horizontally if more than one mode is con-

sidered. For many modes, the energy becomes organized in beams, within which the

ratio approaches the values obtained from the characteristics. The polarization of the

horizontal velocity field also undergoes qualitative changes due to ’non-traditional’ ef-

fects; in particular, the direction of wave propagation can no longer be inferred from

the orientation of the major axis. Finally, the propagation through layers of different

stratification is examined by means of a transmission coefficient. It is shown that the

near-inertial band is special because it is transmitted through layers of any stratifica-

tion. The implications for internal-wave observations are discussed.

1. IntroductionThis paper deals with the properties of internal inertio-gravity waves in

weakly stratified layers, and their propagation through layers of non-uniformstratification. The reason for focussing on weak stratification is that internal-wave properties can be expected to deviate from their usual behaviour becausea commonly made approximation then no longer applies: the so-called ”Tra-ditional Approximation” (TA), in which one neglects the components of theCoriolis force that are proportional to the cosine of latitude. (The familiarCoriolis terms proportional to the sine of latitude are retained under the TA.)This approximation goes back to Laplace, who introduced it in his dynamic the-ory of tides (late 18th century), and it has been widely applied in geophysicalfluid dynamics since. In recent decades, however, its validity has been called intoquestion in a number of areas, such as equatorial dynamics, deep convection,

∗Royal Netherlands Institute for Sea Research (NIOZ), P.O. Box 59, 1790 AB Den Burg,Texel, The Netherlands (gerk@nioz.nl)

†Max Planck Institute for Meteorology, Hamburg, Germany

1

Figure 1: The relative strength of stratification, N/2Ω, derived from temper-ature and salinity profiles in the Pacific Ocean, for a south-north section near179E (WOCE section P14, from the Fiji Islands to the Bering Sea). Note thelogarithmic color scaling. In the deepest layers, the condition N/2Ω À 1 isnot satisfied. Here, non-traditional effects can be expected to significantly af-fect internal-wave dynamics; the same applies to the upper mixed layer. AfterGerkema et al. (2008).

Ekman layers, and internal waves; for a comprehensive review and references,see Gerkema et al. (2008).

Strong stratification tends to suppress ’non-traditional’ effects. This can beunderstood intuitively from the fact that the Coriolis terms proportional to thecosine of latitude (i.e., the non-traditional terms) always involve the verticaldirection: the associated force either is itself vertical, or is induced by verticalvelocity. Since vertical stratification in density tends to reduce vertical move-ments, the effect of the non-traditional Coriolis components, too, is reduced.Generally, non-traditional effects can be neglected if stratification is strong inthe sense that N À 2Ω, where N is the buoyancy frequency, and Ω the Earth’sangular velocity. As a matter of fact, this condition is not satisfied in the deepestlayers of the ocean, as is illustrated in Figure 1.

It has been known for a long time that the range of allowable internal-wavefrequencies is extended by including the non-traditional Coriolis terms (LeBlondand Mysak, 1978). Within this range, three intervals can be distinguished inwhich internal waves behave differently with regard to the horizontal or verticalopposition of energy and phase propagation (van Haren, 2006). This classi-fication is completed here (Section 3). The starting point is the recognitionthat two stratification regimes need to be distinguished, according to which of

2

the two, |f | or (N2 + f2s )1/2, is the largest. (Here f = 2Ω sin φ, with latitude

φ; and fs = 2Ω cos φ sin α, where α is the angle of wave propagation in thehorizontal geographical plane, north of east.) In each regime, the range of al-lowable frequencies includes a near-inertial band, i.e., frequencies around |f |.This means that waves at these frequencies can propagate through layers of anystratification, a phenomenon which is further explored in Section 6 by means ofa transmission coefficient, calculated for a three-layer system.

The linear theory of internal waves can be straightforwardly generalized toinclude non-traditional effects, as was shown by Gerkema and Shrira (2005a).The point is that one ends up with equations that are no more difficult tosolve than under the TA, despite being non-separable in horizontal and verticalcoordinates. In this paper, we extend these results by considering the spatialdistribution of energy density, in particular the ratio of potential and kineticenergies (Section 4), and the polarization of the horizontal velocity field (Section5). In both, we derive the results in terms of characteristic coordinates, underthe assumption of a vertical domain of infinite extension. We compare thiswith results obtained for a vertically bounded domain, using vertical modes.Here, one needs to further distinguish between using a single mode, or moremodes. We demonstrate that the properties thus found are very dfferent fromthose obtained under the TA. Finally, in Section 7, we discuss the implicationsfor the interpretation of observations in the ocean, and the complications non-traditional effects may engender.

2. Basic equations and propertiesWe adopt a coordinate system in which the x-axis is rotated by an angle α

(anti-clockwise) with respect to the west-east direction (Figure 2), and definethe Coriolis parameters as

~f = (fs, fc, f) = 2Ω(sinα cos φ, cosα cos φ, sin φ) . (1)

We assume latitude φ to be fixed (f -plane). Without loss of generality, we takeα to lie between 0 and π; hence we have fs ≥ 0. The components fs and fc areproportional to f (i.e. to cosine of latitude), and would both be absent underthe Traditional Approximation. In this coordinate system, the linear equationsfor an incompressible fluid read

ut − fv + fcw = −px (2)vt + fu− fsw = −py (3)wt + fsv − fcu = −pz + b (4)

ux + vy + wz = 0 (5)bt + N2w = 0 . (6)

The last two equations derive from the principles of mass and energy conserva-tion, respectively. Here u, v and w are the velocity components in the x, y and zdirection, respectively; p is the departure of pressure from its hydrostatic value(divided by a constant reference density, ρ∗); b is the buoyancy, i.e. −gρ/ρ∗,

3

east →

north↑

x

y

α

Figure 2: Coordinate transformation to the x, y-system, at an angle α withrespect to the east-north system. We consider wave propagation along the x-direction, and assume ∂/∂y = 0.

where ρ is the departure of density from its hydrostatic value ρ0(z). Hereafterwe assume N2 = −(g/ρ∗)dρ0/dz to be constant, or, in Section 6, piecewise con-stant. In the remainder of this section we briefly summarize some of the earlierresults (see, e.g. LeBlond and Mysak, 1978; Gerkema and Shrira, 2005a), whichwe need in later sections.

We consider plane waves travelling in the x-direction, so that ∂/∂y = 0,which allows us to introduce a stream function ψ (u = ψz , w = −ψx). We alsoassume that the fields are periodic in time, writing

ψ = Ψ exp(iωt) ,

(the real part being implied), and similarly for the other variables The set (2)–(6) can then be reduced to

AΨxx + 2BΨxz + CΨzz = 0 , (7)

whereA = N2 − ω2 + f2

s ; B = ffs ; C = f2 − ω2 .

Notice that fc is absent from (7), because the terms involving fc cancelled; in(7), ‘non-traditional’ effects are represented by fs alone.

The range of allowable wave frequencies follows from the requirement ofhyperbolicity, i.e. B2 −AC > 0,

ωmin < ω < ωmax ,

with

ωmin,max =1√2

([N2 + f2 + f2

s ]∓

[N2 + f2 + f2s ]2 − (2fN)2

1/2)1/2

. (8)

4

For frequencies that lie within this range, one can define characteristic co-ordinates, ξ± = µ±x− z, with

µ± =B ± (B2 −AC)1/2

A. (9)

Their physical meaning is that wave-energy propagates along lines ξ± =constant,i.e. at steepness µ± in the x, z-plane.

3. Classification of frequency regimes

a. General classificationApart from the fundamentally important frequencies ωmin and ωmax, which

delineate the range of allowable wave frequencies, there are still two other fre-quencies that are special, namely those for which one of the coefficients C or Avanishes:

ω = |f | , ω = (N2 + f2s )1/2 .

The four special frequencies – ωmin, ωmax, |f | and (N2 + f2s )1/2 – satisfy the

following inequalities:

ωmin < min|f |, (N2 + f2

s )1/2

< max|f |, (N2 + f2

s )1/2

< ωmax . (10)

(This follows, with a little algebra, from the expressions for ωmin,max in (8).)Accordingly, we can distinguish two regimes, depending on which of the two, |f |or (N2+f2

s )1/2, is the largest. Thus we define regime I by |f | < (N2+f2s )1/2, and

regime II by (N2 + f2s )1/2 < |f |. In each regime, furtermore, we can subdivide

the range of allowable frequencies into three intervals, as indicated in Table1; in Figure 3 we illustrate how the direction of the characteristics, as well asthe direction of phase and energy propagation, change from one interval to theother.

The presence of a sufficiently strong stratification brings us automatically inregime I, but for weak or absent stratification, either of the regimes may apply,depending on latitude and the direction of wave propagation in the horizontalgeographical plane. So, for example, for propagation in the meridional direction(α = π/2 and fs = 2Ωcos φ), regime II may occur only poleward of 45N/S;equatorward of this latitude, it is precluded (and hence regime I applies), nomatter how weak the stratification. For zonal propagation (α = 0 and fs = 0),on the other hand, regime II always applies in the absence of stratification.

b. Classification under the TAThe TA represents a very special case; the zeros of A and C then coincide

with the lower and upper bounds of the range of allowable frequencies (hencethere is no longer a subdivision into intervals as above), which are now given by|f | and N . Thus, (10) is replaced by the familiar

min|f |, N < ω < max|f |, N (TA) .

5

→

→

→

→

/ / / / / / / / / / /

ξ+

ξ−

ωmin < ω < f

I:

→

→

→

→

/ / / / / / / / / / /

ξ+

ξ−

f < ω <√

N 2 + f 2s

→

→

→

→

/ / / / / / / / / / /

ξ+

ξ−

√

N2 + f2s

< ω < ωmax

→

→

→

→

/ / / / / / / / / / /

ξ+

ξ−

ωmin < ω <√

N2 + f2s

II:

→

→

→

→

/ / / / / / / / / / /

ξ+

ξ−

√

N2 + f2s

< ω < f

→

→→

→

/ / / / / / / / / / /

ξ+

ξ−

f < ω < ωmax

Figure 3: The direction of energy propagation in the x, z-plane, along the char-acteristics ξ+ = const and ξ− = const, for regime I (upper panels) and regimeII (lower panels). They are here depicted as examples of wave reflection from ahorizontal bottom. For each regime, three different subintervals can be distin-guished, and the orientation of one characteristic changes from one interval intothe other. The arrows along the characteristics show the direction of energypropagation; those perpendicular to them, phase propagation. The latter arehere chosen to be to the right. (If taken to be to the left instead, all the arrowswould be reversed.)

6

Parameter regime: Frequency intervals: Characteristics:regime I: ωmin < ω < |f | µ+ > 0 (V) & µ− > 0 (H)|f | < (N2 + f2

s )1/2 |f | < ω < (N2 + f2s )1/2 µ+ > 0 (V) & µ− < 0 (V)

(N2 + f2s )1/2 < ω < ωmax µ+ < 0 (H) & µ− < 0 (V)

regime II: ωmin < ω < (N2 + f2s )1/2 µ+ > 0 (V) & µ− > 0 (H)

(N2 + f2s )1/2 < |f | (N2 + f2

s )1/2 < ω < |f | µ+ < 0 (H) & µ− > 0 (H)|f | < ω < ωmax µ+ < 0 (H) & µ− < 0 (V)

Table 1: The two possible regimes, depending on which of the two, |f | or(N2 + f2

s )1/2, is the largest. Each regime gives rise to three frequency intervals,for which the signs of µ+ and µ− are listed, as well as the corresponding be-haviour of the wavevector and group velocity vector: ‘H’ means that they arehorizontally opposed; ’V’, that they are vertically opposed. This table appliesto the Northern Hemisphere. For the Southern Hemisphere, the signs of µ+, aswell as of µ−, are to be reversed in the first, third, fourth and sixth rows; andin these rows, ‘H’ changes into ‘V’, and vice versa.

Another peculiarity of these bounds is that each depends either on latitude oron stratification, but not on both. None of these properties holds if the TA isabandoned.

c. Propagation through multiple layersThe extension of the range of allowable wave frequencies in the non-traditional

case opens the possibility of propagation through layers of very different strat-ification. This is illustrated in Figure 4. Comparing regimes I and II, we seethat no overlap occurs between the ranges of allowable frequencies if the TAis made (compare dotted lines). Without the TA, however, there is a commonrange of allowable frequencies, namely from ωmin(regime I) to ωmax(regime II).So, internal waves at frequencies within a certain range around |f |, i.e. near-inertial waves, can propagate in both layers, and hence go from one regime intothe other. This subject is elaborated on in Section 6.

4. Ratio of potential and kinetic energiesSolving (7) yields the steamfunction. From this, the other fields are readily

obtained using (2)–(6). Thus, we find for the transverse velocity and buoyancy

V =i

ω(fsΨx + fΨz) (11)

Γ = −iN2

ωΨx , (12)

which have to be multiplied by exp(ıωt) to get v and b, respectively.Kinetic and potential energies, averaged over wave period 2π/ω, can be

written as

Ek = 12ρ∗

⟨<(u)2 + <(v)2 + <(w)2

⟩

7

|ωmin

||f |

||f |

|N

|(N2 + f2

s )1/2

|ωmax

← non−traditional range →

← traditional range →REGIME I

|ωmin

||f |

||f |

|N

|(N2 + f2

s )1/2

|ωmax

← non−traditional range →

← traditional range →REGIME II

Figure 4: Schematic illustration of the ranges of allowable frequencies for thetwo stratification regimes: regime I, where |f | < (N2 + f2

s )1/2, and regime II,where (N2 + f2

s )1/2 < |f |. In this example, the latitude is taken to be the samein both regimes, hence |f | remains the same. However, the stratification N ischosen differently, and ωmin,max change accordingly. For comparison, the rangesunder the TA are shown as well (dotted lines). Along the non-traditional ranges(thick solid lines), the zeros of the coefficients A and C are indicated as well; theysubdivide the non-traditional range into three intervals, a classification whichpertains to the dynamics of internal waves, as indicated in Table 1. (Under theTA, no such subdivision occurs because the zeros of A and C then coincide withthe upper and lower bounds.)

8

= 14ρ∗(UU∗ + V V ∗ + WW ∗) ; (13)

Ep = 12ρ∗

⟨<(b)2/N2

⟩

= 14ρ∗ ΓΓ∗/N2 , (14)

where the brackets 〈· · ·〉 denotes time-averaging, and the asterisk, the complexconjugate.

There are two main roads to solving (7). In the first, characteristic coor-dinates are used; in the second, vertical modes. For both, we derive the keyexpressions for the ratio Ep/Ek, assuming a layer of uniform N ; the differencesbetween the traditional and non-traditional cases are discussed.

a. CharacteristicsThe characteristic coordinates are defined by

ξ± = µ±x− z ,

where µ± is given by (9). Using these coordinates, (7) becomes Ψξ+ξ− = 0,whose general solution is

Ψ = F (ξ+) + G(ξ−) , (15)

for arbitrary functions F and G, each describing propagation of wave-energyalong one of the two characteristics. If we select just one of these functions(F , say) – thus restricting wave propagation to one of the two possible angles –we obtain a simple, space-independent expression for the ratio of potential andkinetic energies

Ep

Ek=

N2µ2+

ω2(1 + µ2+) + (fsµ+ − f)2

. (16)

If G were selected instead of F , the same expression would hold but with µ+

replaced by µ−. The two forms are implicit in, and equivalent to an expressionderived by LeBlond and Mysak (1978), as we show in the appendix.

The form (16) was earlier derived by Gerkema and Shrira (2005b), but, likethe result by LeBlond and Mysak (1978), it was based on the assumption ofplane wave solutions Ψ = exp i(kx+mz +ωt. The present result demonstratesthat (16) is valid for the general solution (15) as well.

Under the TA, we have fs = 0 and µ− = −µ+, so the two characteristicsthen give the same ratio, namely

Ep

Ek=

N2(ω2 − f2)ω2(N2 − f2) + f2(N2 − ω2)

(TA) , (17)

which is a familiar expression (e.g., LeBlond and Mysak, 1978, eq. 8.75).The difference between the non-traditional (15) and the traditional (16) is

illustrated in Figure 5, for three different latitudes. Exactly at the equator, (17)would result in equipartition of potential and kinetic energies (i.e., Ep/Ek = 1)

9

Ep/E

k

φ =5

−

−

1

0 || | |ωmin, |f | N ωmax

Ep/E

k

φ =45

−

−

1

0 | | | |ωmin |f | N ωmax

Ep/E

k

φ =85

−

−

1

0 || ||ωmin, |f | N, ωmax

non−traditional, +non−traditional, −traditional

Figure 5: The ratio of potential and kinetic energies, Ep/Ek, versus wave fre-quency ω. Here a low value of stratification is chosen, N = 4Ω, characteristicof the deepest layer of the ocean (cf. Figure 1). Results are shown for threedifferent latitudes. In each, the ratio for the non-traditional expression (16) isshown, both for µ+ (thick solid line) and for µ− (thick dotted line); here, wavefrequencies must lie between ωmin and ωmax. Also shown is the traditional case(17), thin solid line; here, the frequency is smaller, being constrained by thelower bound |f | and upper bound N .

for all frequencies, but this is not so in the non-traditional case (16), which atthe equator becomes

Ep

Ek=

N2

N2 + 2f2s

. (18)

This value is indicated by the horizontal dotted line in the upper panel of Figure5.

Even though (17) holds irrespective of whether F or G is selected, it doesnot hold if both are involved at the same time. This situation occurs in abasin of finite depth (such as the ocean!), where it is not sufficient to includejust F , because wave reflection from the boundaries (bottom or surface) willresult in propagation along the other characteristic, and hence requires theinclusion of G. Near the surface, or a horizontal bottom, the potential energy

10

must vanish (hence Ep/Ek → 0) because no vertical excursions can occur there.This means (17) is no longer valid. With non-traditional terms included, thesituation becomes even more complicated because the ratios are then differentfor F and G, being given by (16) and by its counterpart with µ−. Either way,in this situation the vertical dependence of Ep/Ek is most easily found in termsof modes, as discussed in the next subsection.

b. Vertical modesIf we consider a vertically confined basin, with a horizontal bottom and

surface, it is more convenient to express the solution directly in terms of verticalmodes, instead of using (15). Thus we write, following Saint-Guily (1970),

Ψ = Ψ(z) exp ik(x−Bz/C) , (19)

substitution of which into (7) gives

d2Ψdz2

+ k2

[B2 −AC

C2

]Ψ = 0 . (20)

Together with the boundary conditions Φ = 0 at z = 0,−H (surface, bottom),this consitutes a Sturm-Liouville problem, which for given latitude, stratifica-tion, and wave frequency yields a discrete set of eigenvalues kn and correspond-ing vertical modes Ψn. Note that Ψn contains only part of the vertical depen-dence, the other part being included in exp(−iknBz/C). (The latter disappearsunder the TA.) The other fields can now be expressed as follows

Un = Ψ′n − iknB

CΨn (21)

Vn = if

ωΨ′n + kn

ωfs

CΨn (22)

Wn = −iknΨn (23)

Γn = N2 kn

ωΨn (24)

Pn =C

ωknΨ′n + fcΨn . (25)

Each has to be multiplied by exp ikn(x−Bz/C) + ωt to get the full solution.We note that the expression for pressure (25) contains fc, demonstrating thatnon-traditional effects would be present even in the case of east-west propaga-tion, when fs = 0.

If we consider only one mode (modenumber n), then the kinetic and potentialenergy is given by

Ek,n = 14ρ∗

[1 +

( f

ω

)2](Ψ′n)2 + k2

n

[B2 + (ωfs)2

C2+ 1

]Ψ2

n

Ep,n = 14ρ∗N2

(kn

ω

)2

Ψ2n . (26)

11

These expressions hold generally, for any profile of N . If we now restrictourselves to the case of constant N , then we can obtain the expression for thedepth-integrated energies. From (20) we have, after multiplication by Ψ andpartial integration,

∫dz (Ψ′n)2 = k2

n

[B2 −AC

C2

] ∫dz Ψ2

n .

After a little algebra, the ratio of the depth-averaged potential and kineticenergies for the uni-modal case is found to be

Ep,n

Ek,n=

N2(ω2 − f2)2

(ω2 − f2)[ω2(N2 − f2) + f2(N2 − ω2)] + 2ω2f2s (ω2 + f2)

. (27)

(The bar stands for depth-averaging.) Under the TA (fs = 0), this reduces sim-ply to (17), but with non-traditional terms included, the expression is differentboth from (16) and from its counterpart with µ−. This means that neither ofthe characteristics produces the ratio for the uni-modal case. At the equator,however, (27) still reduces to (18).

Although the right-hand side of (27) has no dependence on modenumber n,this does not mean that the energy ratio is given by (27) in the multi-modalcase. Under the TA, the modal function Ψn carries the whole vertical depen-dence, and orthogonality of modes Ψn then ensures that products of modes donot contribute to the expressions for the depth-integrated energy. Hence (17)remains valid in the multi-modal case. With fs included, however, there is anextra vertical dependence via the exponential factor exp(−iknBz/C), resultingin contributions of products of modes to the depth-integrated energies. As aconsequence, the ratio Ep/Ek is no longer given by (27); rather, it varies withx.

This is shown in Figure 6. Here, the first ten modes are superposed, resultingin a beam (Figure 6a). The ratio Ep/Ek, shown in Figure 6b, has a layeredstructure, with values alternating between, approximately, zero (white) andone (black). This layering reflects the number of modes involved (there are tenblack layers), and would become finer-grained if more modes were included. Thelayering is, however, most prominent precisely in regions where the total energyis weak, and hence is of little significance. In the region where the beam occurs(cf. Figure 6a) values are more uniform, being low in the descending part ofthe beam, and high in the ascending part. This is further illustrated in Figure6c, where the depth-averaged Ep and Ek are used to form the ratio Ep/Ek

(solid line). Here the connection with the direction of the beam is evident: thevariation in x reflects precisely the transition from one characteristic to another.The alternation between high and low values in Figure 6c, as indicated by thesolid line, does not yet correspond to the values for the characteristics givenby (16) and its counterpart with µ− (indicated by the horizontal dotted lines).However, numerical evidence demonstrates that the latter values are approachedif more modes are taken into account; an example is provided by the dashedline in Figure 6c, which is based on 100 modes.

12

Interestingly, the expression for the uni-modal case (27), too, remains mean-ingful, for if we take not only the vertical average of Ep and Ek, but also thehorizontal average, we obtain a ratio E′

p/E′k which is always equal to (27), no

matter how many modes are involved. No such horizontal averaging would beneeded under the TA, of course, because in that case the ratio Ep/Ek itself isalready horizontally uniform.

5. Polarization of the horizontal velocity fieldWe now consider the polarization of the horizontal velocity field, again ex-

amining the methods of characteristic and vertical modes separately.

a. CharacteristicsWith the streamfunction Ψ again given by (15), we obtain U from Ψz, and

V from (11). Recall that U is the current velocity in the direction of wavepropagation, which is at an angle α north of east (Figure 2).

Selecting again F , we find for u and v:

u = −<F ′ exp(iωt) ; v = ω−1(fsµ+ − f)<iF ′ exp(iωt) ,

where the prime denotes the derivative to the argument of F , i.e. F ′ = ∂F/∂ξ+.These expressions imply that u and v have a phase difference of 90. The sameargument applies if G were selected instead of F , in which case we have toreplace µ+ by µ−. The ratio of the major and minor axes of the u, v-hodographis therefore simply given by

|v|/|u| = |fsµ± − f |/ω . (28)

There are some special frequencies that require attention. First, the inertialfrequency ω = |f |, in which case µ− = 0, so that polarization is circular. Thisis, however, not the case for the other characteristic, µ+. Second, ω = (N2 +f2

s )1/2, i.e. A = 0, which implies that µ± becomes infinite, resulting in rectilinearpolarization, in a direction perpendicular to wave propagation. Finally, ω =N , in which case we have µ± = (f ± ω)/fs, hence (28) now implies circularpolarization. This holds for any N , no matter how large, and thus poses aparadox because one would not expect high-frequency waves to be affected byCoriolis effects. This paradox can be resolved in two ways. First, we havesupposed that waves exist ‘eternally’ by assuming exp iωt. In reality, high-frequency waves are dissipated long before Coriolis effects could have made anyimpact (which requires time scales of the order of Ω−1). Second, even if waveswere to live long enough for the horizontal velocity field to become circular,this would still be insignificant because at high frequencies ω ≈ N À Ω theexcursion of the water parcels is predominantly vertical anyway. Still, the caseω = N provides an interesting example of the singular nature of the TA: as longas the TA is not made, the horizontal polarization is circular, even for strongstratification; whereas if one makes the TA, it becomes nearly rectilinear at highfrequencies.

As noted, (28) applies if either F or G is selected. However, if both F and Gare involved at the same time, then the phase difference between u and v need

13

z(k

m)

a) Ep + Ek

0 5 10 15 20 25 30 35−4

−3

−2

−1

0

0

0.1

0.2

z(k

m)

b) Ep/Ek

0 5 10 15 20 25 30 35−4

−3

−2

−1

0

0

0.25

0.5

0.75

1

0 5 10 15 20 25 30 350

0.25

0.5

0.75

1c)

x (km)

Ep/Ek

Figure 6: A multi-modal example, with ten modes included, showing a) thelocation of the beam, where the energy is concentrated (total energy density,Ep+Ek, is here normalized arbitrarily). In b) the ratio Ep/Ek is shown; it variesboth with x and with z. In c) the ratio of the depth-averaged potential andkinetic energies, Ep and Ek, which varies with x; high (low) values are foundwhere the beam follows the direction of the µ+ (µ−) characteristic. The levelscorresponding to these characteristics are indicated by the two horizontal dottedlines; the upper line is based on (16), the lower line on its counterpart with µ−.These levels are approached if many more modes are taken into account; anexample is shown for 100 modes (dashed line). Parameters are: N = 4Ω,φ = 45N, σ = 1.5× |f |, and α = π/2 (northward propagation).

14

no longer be 90, and the ellipticity can no longer be expressed simply as theratio between the two. (This point is demonstrated in the following section.)Under the TA, on the other hand, the phase difference is always 90, and theratio of the major and minor axes is always |f |/ω.

In other words, the simplicity of (28) is deceptive, because it hinges on theselection of just one of the functions F or G, which is possible only in a mediumof infinite extension. We elaborate on this point in the following section, wherewe consider a layer of finite vertical extension and use vertical modes.

b. Vertical modesWe now turn to the polarization of the horizontal velocity field in terms of

modes. Writingu = <(U) cos ωt−=(U) sin ωt ,

and similarly for v, we find that

u2 + v2 = 12 [<(U)2 + <(V )2 + =(U)2 + =(V )2]

+ 12 cos(2ωt)[<(U)2 + <(V )2 −=(U)2 −=(V )2]

− sin(2ωt)[<(U)=(U) + <(V )=(V )] , (29)

which can be rewritten as

u2 + v2 = a + b cos(χ + 2ωt) ,

with

a = 12 [<(U)2 + <(V )2 + =(U)2 + =(V )2] (30)

b = (c21 + c2

2)1/2 (31)

c1 = 12 [<(U)2 + <(V )2 −=(U)2 −=(V )2] (32)

c2 = <(U)=(U) + <(V )=(V ) (33)

and cos χ = c1/b, sinχ = c2/b.This implies that the radius of the major axis of the horizontal current

field is (a + b)1/2, and the radius of the minor axis (a − b)1/2; we define thepolarization as the ratio between the two. The major axis, moreover, is reachedwhen t = t∗ = −χ/(2ω). We thus define the orientation of the ellipse, i.e. theangle in the horizontal plane between the major axis and the x-direction, byarctan(v(t∗)/u(t∗)).

We first briefly discuss the results obtained under the TA, which are trivial.The polarization is then always given by |f |/ω; this follows from (3) with fs = 0and ∂/∂y = 0. Furthermore, the orientation of the ellipse is either 0 or 90,depending on which of the two, |f | or ω, is the largest. This, in turn, dependson the stratification: if N < |f |, then ω < |f |; and if N > |f |, then ω > |f | (seeFigure 4).

With non-traditional effects included, the situation becomes more intricate.We consider first, in Figure 7, the case with the same parameters as in Figure 6;

15

here, too, ten modes are included. In Figure 7a we recognize the beam structurefrom Figure 6a, now depicted in terms of the magnitude of the major axis. Fig-ure 7b demonstrates that the polarization varies spatially; values are smaller inthe upward than in the downward beam. Within these beams, the polarizationis on average close to the values derived for the characteristics, which are givenby (28), yielding 0.87 for µ−, and 0.26 for µ+. As one would expect, they comecloser to the latter values if more modes are included (not shown). If, instead,we use fewer modes, the reverse happens (Figure 8b). With just two modes, ofcourse, we can no longer speak of a beam. The most surprising outcome is whathappens if we take a single mode only. The polarization then becomes spatiallyuniform, as under the TA; moreover, it even takes the same value as under theTA (Figure 9b). Numerically, it is found that this happens for any stratificationand for any mode n (as long as we restrict ourselves to a single mode); for onemode, the polarization is always given by the ‘traditional’ value |f |/ω. (Moreillustrations of this result are included in Exarchou (2007).)

However, even for one mode there are still important departures from thetraditional result. Under the TA, the first mode would have vanishing u andv halfway down the water column, which clearly is not the case in the non-traditional Figure 9a. The reason simply lies in the extra (i.e. second) z depen-dence in (19), which produces the second terms on the right in (21) and (22).Another, more surprising departure concerns the orientation of the ellipse. Eventhough the ratio of major to minor axis is in the uni-modal case the same asunder the TA, the orientation of the major axis is different, and, moreover,varies vertically, covering all possible angles (Figure 9c). In other words, non-traditional effects create a tilt of the ellipse. If more modes are included, as inFigure 8c, the orientation varies horizontally as well, but tends to become uni-form (in this case: zero) within the beam if many modes are involved (Figures7c).

To summarize, we find that for one mode the ratio of major to minor axisis uniform and takes the traditional value, but, unlike under the TA, the majoraxis has a tilt that varies vertically. For many modes, the tilt tends to disappearwithin the beam, but the ratio of major to minor axis approaches the valuescorresponding to the characteristics, which, unlike under the TA, are differentfor the upward and downward beams.

6. Propagation through layers of different stratificationWe now consider the propagation through layers of different stratification

N . Specifically, we adopt a three-layer system, in which the upper and lowerlayer are of semi-infinite extension, while the middle layer has a thickness h.The upper and lower layer have N(z) = N1; and the middle layer, N(z) = N2,where N1,2 are constants (Figure 10). We want to determine how much of theincoming energy (from below, say) is transmitted through the middle layer, andhow this depends on the parameters of the system, and on making the TA ornot.

To solve this problem, we return to (7), in which A now depends on z,because N varies with z. However, within each layer, A is constant, and for

16

z(k

m)

a) radius major axis

0 5 10 15 20 25 30 35−4

−3

−2

−1

0

0

0.2

0.4

0.6

0.8

1

z(k

m)

b) ratio minor and major axes

0 5 10 15 20 25 30 35−4

−3

−2

−1

0

0

0.2

0.4

0.6

0.8

1

x (km)

z(k

m)

c) orientation major axis

0 5 10 15 20 25 30 35−4

−3

−2

−1

0

−90−60−300306090

Figure 7: A multi-modal example, with ten modes included, showing propertiesof the horizontal velocity field (u, v). In a) the location of the beam; in b) thepolarization; and in c) the orientation of the major axis in the horizontal plane.Parameters are as in Figure 6.

17

z(k

m)

a) radius major axis

0 5 10 15 20 25 30 35−4

−3

−2

−1

0

0

0.2

0.4

0.6

0.8

1

z(k

m)

b) ratio minor and major axes

0 5 10 15 20 25 30 35−4

−3

−2

−1

0

0

0.2

0.4

0.6

0.8

1

x (km)

z(k

m)

c) orientation major axis

0 5 10 15 20 25 30 35−4

−3

−2

−1

0

−90−60−300306090

Figure 8: As in Figure 7, but now with only the first two modes.

18

z(k

m)

a) radius major axis

0 5 10 15 20 25 30 35−4

−3

−2

−1

0

0

0.2

0.4

0.6

0.8

1

z(k

m)

b) ratio minor and major axes

0 5 10 15 20 25 30 35−4

−3

−2

−1

0

0

0.2

0.4

0.6

0.8

1

x (km)

z(k

m)

c) orientation major axis

0 5 10 15 20 25 30 35−4

−3

−2

−1

0

−90−60−300306090

Figure 9: As in Figure 7, but now for the first mode alone.

19

N1

N2

N →−h/2

−−h/2

z ↑

Figure 10: A three layer system, consisting of semi-infinite upper and lowerlayers with N(z) = N1, and a middle layer of thickness h and stratificationN(z) = N2. Waves enter the system from below.

later reference we define the constants

A1,2 = N21,2 − ω2 + f2

s .

Via the transformation (19), we reduce (7) to (20),

d2Ψdz2

+ k2

[B2 −AC

C2

]Ψ = 0 . (34)

Within each layer, the part in square brackets is constant, so the solution iseither sinusoidal or exponential, depending on the sign of B2−AC. We assumethat waves are sinusoidal in the upper and lower layers, i.e. wave frequenciesmust be such that B2 − A1C > 0. For the middle layer, we assume for themoment that the solution is exponential, i.e. B2 − A2C < 0. The solution of(34) can thus be written

Ψ =

c1eiqz + c2e

−iqz (z < −h/2 , lower layer)c3e

κz + c4e−κz (−h/2 < z < h/2 , middle layer)

c5eiqz + c6e

−iqz (z > h/2 , upper layer) ,(35)

where

q = k∆1/2

1

C; κ = k

(−∆2)1/2

C,

with q and κ both real, and ∆1,2 = B2 −A1,2C.At the levels z = ±h, we have to require continuity of Ψ and of its derivative

Ψ′, thus guaranteeing continuity of the vertical velocity and pressure, given by(23) and (25), respectively. This allows us to express the coefficients c1,2 interms of c5,6. Assuming waves to enter from below, we may take c6 = 0. Theproblem is now mathematically identical to that of a rectangular potential bar-rier in quantum mechanics (Merzbacher, 1998, §6.2), which yields the following

20

expression for the transmission coefficient T = |c5|2/|c1|2,

T =1

cosh2 κh− (∆1+∆2)2

4∆1∆2sinh2 κh

=1

cosh2[kh(−∆2)1/2/C]− (∆1+∆2)2

4∆1∆2sinh2[kh(−∆2)1/2/C]

. (36)

An expression equivalent to (36) was used before by Sutherland and Yewchuk(2004), but there ∆1,2 were defined differently because Coriolis effects wereignored altogether. Here we include them and examine the effect of the TA.

So far we have assumed that Ψ is exponential in the middle layer (i.e. ∆2 < 0and κ real). Eq. (36) applies however also when ∆2 > 0, in which case we canwrite

T =1

cos2[kh∆1/22 /C] + (∆1+∆2)2

4∆1∆2sin2[kh∆1/2

2 /C]. (37)

Thus, both regimes of the middle layer can be considered. For given latitudeand stratification, the transmission coefficient T in (36) and (37) depends ontwo parameters: on wave frequency ω (via ∆1,2 and C), and on kh, the thicknessof the middle layer, measured against the horizontal wavelength.

In the figures that follow, we plot T as a function of these two parameters.In each figure, we take N1 = 4Ω, φ = 45N and α = π/2 (propagation in themeridional direction). Only N2, the stratification of the middle layer, is varied:we take N2 = 0 (i.e., a convective layer) in Figure 11, N2 = 2Ω in Figure 12,and N2 = 6Ω in Figure 13. In each, the non-traditional result is shown in theupper panel; the traditional result, in the lower panel.

With a convective middle layer (Figure 11), the contrast is starkest. Underthe TA, waves are evanescent in the middle layer at all frequencies, and as aresult, little energy passes through it, except for very thin layers (kh ¿ 1).Without the TA, by contrast, energy at low frequencies, i.e. near f , passesthe middle layer virtually unhindered, regardless of the thickness of the middlelayer. In Figure 11a, the transition between wave frequencies for which ∆2 > 0(propagation in the middle layer), on the left, and those for which ∆2 > 0(evanescence), on the right, is clearly seen (and indicated by an arrow). Evenwhen waves can propagate through the middle layer, not all the incoming energypasses through it, because internal reflections occur at the levels separating thelayers, where N changes from one value to another. The intensity of the internalreflections depends on the thickness of the middle layer, as Figure 11a shows.

With a stronger stratification in the middle layer (Figure 12), there is somewave propagation through the middle layer at low frequencies under the TA,but without the TA, the transmission is higher at all frequencies, especially soat near-inertial ones. This is still true when the stratification of the middle layerexceeds that of the outer layers (Figure 13). In this case, wave can propagatethrough the middle layer at all frequencies, except the very lowest in Figure13a. Still, there is no full transmission at higher frequencies, because of internalreflections, as noted above.

21

ω/f

kh

a) T (non-traditional)

↑1 2 30

1

2

3

4

5

0

0.2

0.4

0.6

0.8

1

ω/f

kh

b) T (traditional)

1 2 30

1

2

3

4

5

0

0.2

0.4

0.6

0.8

1

Figure 11: The transmission coefficient T as a function of scaled wave frequency,ω/f , and of scaled middle-layer thickness, kh. Latitude is φ = 45N and theouter layers have stratification N1 = 4Ω. Here, the middle layer is convective:N2 = 0. The upper panel shows the non-traditional result; the horizontal axiscovers exactly the non-traditional frequency range (ωmin, ωmax) that is basedon N1 (outer layers). The arrow indicates the transition from oscillatory toevanescent behaviour in the middle layer. Under the TA, the frequency rangeis of course smaller, and runs from f to N1, as indicated by the vertical dashedlines in the lower panel; here, waves are evanescent throughout.

22

ω/f

kh

a) T (non-traditional)

↑1 2 30

1

2

3

4

5

0

0.2

0.4

0.6

0.8

1

ω/f

kh

b) T (traditional)

↑1 2 30

1

2

3

4

5

0

0.2

0.4

0.6

0.8

1

Figure 12: As in Figure 11, but now with weak stratification in the middle layer:N = 2Ω.

23

ω/f

kh

a) T (non-traditional)

1 2 30

1

2

3

4

5

0

0.2

0.4

0.6

0.8

1

ω/f

kh

b) T (traditional)

1 2 30

1

2

3

4

5

0

0.2

0.4

0.6

0.8

1

Figure 13: As in Figure 11, but with a middle-layer stratification exceeding thatof the outer layers: N2 = 6Ω.

24

7. ConclusionIt was argued by Garrett (2001) that the near-inertial band stands out in the

internal-wave spectrum. Not only is it the most energetic part of the spectrum,but also the propapagation is different; near-inertial waves undergo, as it were, ashift within the band when they propagate in the meridional direction, becausethe meaning of “inertial” changes with latitude. Here we have analysed anotheraspect of meridional propagation, namely the passage through layers of differentstratification (Section 6). Here, too, the near-inertial band is special becauseit is the only part of the spectrum that can pass layers of any stratification.This means also that meridional propagation is favoured, because no such non-traditional effects do occur for zonal propagation.

This is especially remarkable in the case of convective layers (with N zero)beneath or above a layer of significant stratification. Under the TA, no fre-quencies would exist for which waves can propagate in both layers, but withnon-traditional effects included, this becomes possible for near-inertial waves.Observational evidence by van Haren and Millot (2004) confirms the latter: inthe Mediterranean Sea, near-inertial were observed in both kinds of layers atthe same time. They also found that the polarization of the horizontal veloc-ity was very different in the convective layer; instead of being circular it wasellipse-like. As noted before (Gerkema et al., 2008), this is in accordance withnon-traditional theory, because the expression (28) in terms of the µ+ charac-teristic (which is associated with vertical energy propagation, the other one, µ−giving horizontal propagation at the inertial frequency) predicts such a depar-ture from circular polarization. However, the analysis of Section 5 shows thatthe situation is actually more complicated if we take in account the verticalconfinement of the basin. The result is then found to depend strongly on thenumber of modes; for one mode, polarization would be circular at the inertialfrequency; it would become an ellipse only if more modes were included, butthen still varies strongly spatially. Yet, the non-traditional theory is confirmedbecause under the TA the polarization would always be circular at the inertialfrequency. We also note the complication of the tilt of the ellipse, engenderedby non-traditional effects, which implies that the direction of wave propagationcan no longer be inferred from the orientation of the major axis.

A final remark concerns our findings on the ratio of potential and kineticenergies. This ratio depends strongly on the number of modes, and varies bothhorizontally and vertically if more than one mode is involved. Moreover, thereis a significant departure from the ratio as obtained under the TA (see, e.g.,Figure 5). Apart from being of fundamental interest, this modification shouldalso be relevant to applications in which the expression for internal-wave en-ergy features. A case in point is the parameterization used to calculate verticaleddy diffusivity from CTD and LADCP profiles (e.g., Kunze et al., 2006). Thisparameterization is likely to require modification in weakly stratified layers, aswell as in the equatorial region.

Acknowledgements. We thank Victor Shrira, Hans van Haren and Leo Maasfor inspiring discussions on the subject matter of this paper.

25

APPENDIX

A. Ratio of potential and kinetic energiesFor plane waves, LeBlond and Mysak (1978) derived an expression for the

ratio of potential and kinetic energies, with non-traditional terms included,namely:

Ep

Ek=

N2 cos2 θ

ω2 + 4Ω2 cos2 α′(38)

(their equation 8.65). Here θ denotes the angle of the wavevector ~k with thehorizontal plane, and α′ the angle between the wavevector and the rotationvector ~Ω. We here demonstrate that this expression is equivalent to our (16).

We start with the identity cot θ = −µ± (the dispersion relation in implicitform), which was derived by Gerkema and Shrira (2005a). Hence cos θ =−µ±/(1 + µ2

±)1/2. Using the inner product, we can write ~k · ~Ω = |~k|Ωcos α′.With ~k = |~k|(cos θ, 0, sin θ) and ~Ω = ~f/2, with ~f defined by (1), we then find

cosα′ = cos θ sin α cos φ + sin θ sin φ =−µ±fs + f

2Ω(1 + µ±)1/2.

Hence, (38) can also be written as

Ep

Ek=

N2µ2±

ω2(1 + µ2±) + (fsµ± − f)2

, (39)

which is precisely (16), and its counterpart with µ−.

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