Internal-wave properties in weakly stratiÞed layers · vertically bounded domain, using vertical modes. Here, one needs to further distinguish between using a single mode, or more

Journal of Marine Research 66 617ndash644 2008

Internal-wave properties in weakly stratified layers

by Theo Gerkema1 and Eleftheria Exarchou2

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 latitude (which are neglected in the so-calledTraditional Approximation) A systematic overview is given of the wave properties if these terms areincluded The usage of characteristics and vertical modes is compared and contrasted It is shown thatthe ratio of potential and kinetic energy varies horizontally if more than one mode is considered Formany modes the energy becomes organized in beams within which the ratio approaches the valuesobtained from the characteristics The polarization of the horizontal velocity field also undergoesqualitative changes due to lsquonon-traditionalrsquo effects In particular the horizontal direction of wavepropagation can no longer be inferred from the orientation of the major axis of the ellipse also thepolarization may now become counterclockwise in the Northern Hemisphere Finally the propagationthrough layers of different stratification is examined by means of a transmission coefficient It is shownthat the near-inertial band is special because it is transmitted through layers of any stratification Theimplications for internal-wave observations are discussed

1 Introduction

This paper deals with the properties of internal inertio-gravity waves in weakly stratifiedlayers and their propagation through layers of non-uniform stratification The reason forfocussing on weak stratification is that internal-wave properties can be expected to deviatefrom their usual behaviour because a commonly made approximation then no longer appliesthe so-called ldquoTraditional Approximationrdquo (TA) in which one neglects the components ofthe Coriolis force that are proportional to the cosine of latitude (The familiar Coriolisterms proportional to the sine of latitude are retained under the TA) This approximationgoes back to Laplace who introduced it in his dynamic theory of tides (late 18th century)and it has been widely applied in geophysical fluid dynamics ever since In recent decadeshowever its validity has been called into question in a number of areas such as equatorialdynamics deep convection Ekman layers and internal waves for a comprehensive reviewand references see Gerkema et al (2008)

1 Royal Netherlands Institute for Sea Research (NIOZ) PO Box 59 1790 AB Den Burg Texel The Nether-lands email gerknioznl

2 Max Planck Institute for Meteorology Hamburg Germany

617

618 Journal of Marine Research [66 5

Strong stratification tends to suppress lsquonon-traditionalrsquo effects This can be understoodintuitively from the fact that the Coriolis terms proportional to the cosine of latitude (iethe non-traditional terms) always involve the vertical direction the associated force either isitself vertical or is induced by vertical velocity Since stable vertical stratification in densitytends to reduce vertical movements the effect of the non-traditional Coriolis componentstoo is reduced Generally non-traditional effects can be neglected if stratification is strongin the sense that N 2Ω where N is the buoyancy frequency and Ω the Earthrsquos angularvelocity As a matter of fact this condition is not satisfied in the deepest layers of the oceanas is illustrated in Figure 1

It has been known for a long time that the range of allowable internal-wave frequenciesis extended by including the non-traditional Coriolis terms (LeBlond and Mysak 1978)Within this range three intervals can be distinguished in which internal waves behave dif-ferently with regard to the horizontal or vertical opposition of energy and phase propagation(van Haren 2006) This classification is completed here (Section 3) The starting point isthe recognition that two stratification regimes need to be distinguished according to whichof the two |f | or (N2 + f 2

s )12 is the largest (Here f = 2Ω sin φ with latitude φ andfs = 2Ω cos φ sin α where α is the angle of wave propagation in the horizontal geograph-ical plane north of east) In each regime the range of allowable frequencies includes a

Figure 1 The relative strength of stratification N2Ω derived from temperature and salinity profilesin the Pacific Ocean for a south-north section near 179E (WOCE section P14 from the Fiji Islandsto the Bering Sea) Here N is calculated using its definition N2 = g2(partρ0partp minus cminus2

s ) where ρ0is in-situ density p is pressure and cs the speeed of sound Note the logarithmic gray scaling Inthe deepest layers the condition N2Ω 1 is not satisfied Here non-traditional effects can beexpected to significantly affect internal-wave dynamics the same applies to the upper mixed layerAfter Gerkema et al (2008)

2008] Gerkema amp Exarchou Internal-wave properties in weakly stratified layers 619

near-inertial band ie frequencies around |f | This means that waves at these frequenciescan propagate through layers of any stratification a phenomenon which is further exploredin Section 6 by means of a transmission coefficient calculated for a three-layer system

The linear theory of internal waves can be straightforwardly generalized to include non-traditional effects as was shown by Gerkema and Shrira (2005a) The point is that one endsup with equations that are no more difficult to solve than under the TA despite being non-separable in horizontal and vertical coordinates In this paper we extend these results byconsidering the spatial distribution of energy density in particular the ratio of potential andkinetic energies (Section 4) and the polarization of the horizontal velocity field (Section 5)For both we derive the results in terms of characteristic coordinates under the assumptionof a vertical domain of infinite extension We compare this with results obtained for avertically bounded domain using vertical modes Here one needs to further distinguishbetween using a single mode or more modes We demonstrate that the properties thusfound are very dfferent from those obtained under the TA Finally in Section 7 we discussthe implications for the interpretation of observations in the ocean and the complicationsnon-traditional effects may engender

2 Basic equations and properties

We adopt a coordinate system in which the x-axis is rotated at an angle α (anti-clockwise)with respect to the west-east direction (Fig 2) and define the Coriolis parameters as

f = (fs fc f ) = 2Ω(sin α cos φ cos α cos φ sin φ) (1)

Figure 2 Coordinate transformation to the x y-system at an angle α with respect to the east-northsystem We consider wave propagation along the x-direction and assume partparty = 0


with Ω = 72921times10minus5 rads We assume latitude φ to be fixed Without loss of generalitywe take α to lie between 0 and π hence we have fs ge 0 The components fs and fc

are proportional to f = 2Ω cos φ (ie to cosine of latitude) and would both be absentunder the Traditional Approximation In this coordinate system the linear equations for anincompressible fluid read

ut minus f v + fcw = minuspx (2)

vt + f u minus fsw = minuspy (3)

wt + fsv minus fcu = minuspz + b (4)

ux + vy + wz = 0 (5)

bt + N2w = 0 (6)

The last two equations derive from the principles of mass and energy conservation respec-tively Here u v and w are the velocity components in the x y and z direction respectivelyp is the departure of pressure from its hydrostatic value (divided by a constant referencedensity ρlowast) b is the buoyancy ie minusgρρlowast where ρ is the departure of density from itshydrostatic value ρ0(z) Hereafter we assume N to be constant or in Section 6 piecewiseconstant In the remainder of this section we briefly summarize some of the earlier results(see eg LeBlond and Mysak 1978 Gerkema and Shrira 2005a) that we need in latersections

We consider plane waves travelling in the x-direction so that partparty = 0 which allowsus to introduce a stream function ψ (u = ψz w = minusψx) We also assume that the fieldsare periodic in time writing

ψ = Ψ exp(iωt)

(the real part being implied) and similarly for the other variables The set (2)ndash(6) can thenbe reduced to

AΨxx + 2BΨxz + CΨzz = 0 (7)

where

A = N2 minus ω2 + f 2s B = ffs C = f 2 minus ω2

Notice that fc is absent from (7) because the terms involving fc cancelled in (7) lsquonon-traditionalrsquo effects are represented by fs alone

The range of allowable wave frequencies follows from the requirement of hyperbolicityie B2 minus AC gt 0

ωmin lt ω lt ωmax

with

ωminmax = 1radic2

([N2 + f 2 + f 2

s

] ∓ [N2 + f 2 + f 2

s

]2 minus (2f N)212)12 (8)


For frequencies that lie within this range one can define characteristic coordinates ξplusmn =microplusmnx minus z with

microplusmn = B plusmn (B2 minus AC)12

A (9)

Their physical meaning is that wave-energy propagates along lines ξplusmn = constant ie atsteepness microplusmn in the x z-plane

3 Classification of frequency regimes

a General classification

Apart from the fundamentally important frequencies ωmin and ωmax which delineate therange of allowable wave frequencies there are still two other frequencies that are specialnamely those for which one of the coefficients C or A vanishes

ω = |f | ω = (N2 + f 2

s

)12

The four special frequencies ndash ωmin ωmax |f | and (N2 + f 2s )12 ndash satisfy the following

inequalities

ωmin lt min|f | (N2 + f 2

s

)12lt max

|f | (N2 + f 2s

)12lt ωmax (10)

(This follows with a little algebra from the expressions for ωminmax in (8)) Accordinglywe can distinguish two regimes depending on which of the two |f | or (N2 +f 2

s )12 is thelargest Thus we define regime I by |f | lt (N2+f 2

s )12 and regime II by (N2+f 2s )12 lt |f |

In each regime furthermore we can subdivide the range of allowable frequencies intothree intervals as indicated in Table 1 in Figure 3 we illustrate how the direction of thecharacteristics as well as the direction of phase and energy propagation change from oneinterval to the other

The presence of a sufficiently strong stratification brings us automatically in regimeI but for weak or absent stratification either of the regimes may apply depending onlatitude and the direction of wave propagation in the horizontal geographical plane Sofor example for propagation in the meridional direction (α = π2 and fs = 2Ω cos φ)regime II may occur only poleward of 45NS equatorward of this latitude it is precluded(and hence regime I applies) no matter how weak the stratification For zonal propaga-tion (α = 0 and fs = 0) on the other hand regime II always applies in the absence ofstratification

When N = 0 the sole restoring force is the Coriolis force internal waves are then alsoknown as lsquogyroscopicrsquo or (somewhat confusingly) lsquoinertialrsquo waves The bounds of the rangeof allowable frequencies is then given by ωmin = 0 and ωmax = (f 2 + f 2

s )12 and wavespropagate as indicated in the upper or lower panels of Figure 3 depending on which of


Table 1 The two possible regimes depending on which of the two |f | or (N2 +f 2s )12 is the largest

Each regime gives rise to three frequency intervals for which the signs of micro+ and microminus are listedas well as the corresponding behaviour of the wavevector and group velocity vector lsquoHrsquo meansthat they are horizontally opposed lsquoVrsquo that they are vertically opposed This table applies to theNorthern Hemisphere For the Southern Hemisphere the signs of micro+ as well as of microminus are to bereversed in the first third fourth and sixth rows and in these rows lsquoHrsquo changes into lsquoVrsquo and viceversa

Parameter regime Frequency intervals Characteristics

regime I ωmin lt ω lt |f | micro+ gt 0 (V) amp microminus gt 0 (H)|f | lt (N2 + f 2

s )12 |f | lt ω lt (N2 + f 2s )12 micro+ gt 0 (V) amp microminus lt 0 (V)

(N2 + f 2s )12 lt ω lt ωmax micro+ lt 0 (H) amp microminus lt 0 (V)

regime II ωmin lt ω lt (N2 + f 2s )12 micro+ gt 0 (V) amp microminus gt 0 (H)

(N2 + f 2s )12 lt |f | (N2 + f 2

s )12 lt ω lt |f | micro+ lt 0 (H) amp microminus gt 0 (H)|f | lt ω lt ωmax micro+ lt 0 (H) amp microminus lt 0 (V)

Figure 3 The direction of energy propagation in the x z-plane along the characteristics ξ+ = const

and ξminus = const for regime I (upper panels) and regime II (lower panels) They are here depicted asexamples of wave reflection from a horizontal bottom For each regime three different subintervalscan be distinguished and the orientation of one characteristic changes from one interval intothe other The arrows along the characteristics show the direction of energy propagation thoseperpendicular to them phase propagation The latter are here chosen to be to the right (If taken tobe to the left instead all the arrows would be reversed)


the two regimes applies which in turn depends on latitude and the horizontal direction ofwave propagation (ie α) as pointed out above

b Classification under the TA

The TA represents a very special case the zeros of A and C then coincide with thelower and upper bounds of the range of allowable frequencies (hence there is no longera subdivision into intervals as above) which are now given by |f | and N Thus (10) isreplaced by the familiar

min|f | N lt ω lt max|f | N (TA)

Another peculiarity of these bounds is that each depends either on latitude or on stratificationbut not on both None of these properties holds if the TA is abandoned

As in the non-traditional case a range of allowable frequencies exists if N lt |f | withN lt ω lt |f | even though the range is smaller than without the TA In this regime ofsub-inertial internal waves the restoring forces are gravity and the Coriolis force just aswith the more familiar super-inertial internal waves In the limiting case of an unstratifiedlayer only the Coriolis force remains as the restoring force in which case we speak ofgyroscopic waves as mentioned above

c Propagation through multiple layers

The extension of the range of allowable wave frequencies in the non-traditional caseopens the possibility of propagation through layers of very different stratification This isillustrated in Figure 4 Comparing regimes I and II we see that no overlap occurs between theranges of allowable frequencies if the TA is made (compare dotted lines) Without the TAhowever there is a common range of allowable frequencies namely from ωmin(regime I)to ωmax(regime II) So internal waves at frequencies within a certain range around |f | ienear-inertial waves can propagate in both layers and hence go from one regime into theother This subject is elaborated on in Section 6

4 Ratio of potential and kinetic energies

Solving (7) yields the steamfunction From this the other fields are readily obtained using(2)ndash(6) Thus we find for the transverse velocity and buoyancy

V = i

ω(fsΨx + f Ψz) (11)

Γ = minusiN2

ωΨx (12)

which have to be multiplied by exp(iωt) to get v and b respectively


Figure 4 Schematic illustration of the ranges of allowable frequencies for the two stratificationregimes regime I where |f | lt (N2 + f 2

s )12 and regime II where (N2 + f 2s )12 lt |f | In this

example the latitude is taken to be the same in both regimes hence |f | remains the same Howeverthe stratification N is chosen differently and ωminmax change accordingly For comparison theranges under the TA are shown as well (dotted lines) Along the non-traditional ranges (thick solidlines) the zeros of the coefficients A and C are indicated as well they subdivide the non-traditionalrange into three intervals a classification which pertains to the dynamics of internal waves asindicated in Table 1 (Under the TA no such subdivision occurs because the zeros of A and C thencoincide with the upper and lower bounds)

Kinetic and potential energies averaged over wave period 2πω can be written as

Ek = 1

2ρlowast〈(u)2 + (v)2 + (w)2〉

= 1

4ρlowast(UUlowast + V V lowast + WW lowast) (13)

Ep = 1

2ρlowast〈(b)2N2〉

= 1

4ρlowastΓΓlowastN2 (14)

where the brackets 〈middot middot middot 〉 denotes time-averaging and the asterisk the complex conjugateThere are two main roads to solving (7) In the first characteristic coordinates are used

in the second vertical modes For both we derive the key expressions for the ratio EpEk


assuming a layer of uniform N the differences between the traditional and non-traditionalcases are discussed

a Characteristics

The characteristic coordinates are defined by ξplusmn = microplusmnx minus z where microplusmn is given by (9)Using these coordinates (7) becomes Ψξ+ξminus = 0 whose general solution is

Ψ = F(ξ+) + G(ξminus) (15)

for arbitrary functions F and G each describing propagation of wave-energy along one ofthe two characteristics If we select just one of these functions (F say) ndash thus restrictingwave propagation to one of the two possible angles ndash we obtain a simple space-independentexpression for the ratio of potential and kinetic energies

Ep

Ek

= N2micro2+

ω2(1 + micro2

+) + (fsmicro+ minus f )2

(16)

If G were selected instead of F the same expression would hold but with micro+ replaced bymicrominus The two forms are implicit in and equivalent to an expression derived by LeBlond andMysak (1978) as we show in the appendix

The form (16) was earlier derived by Gerkema and Shrira (2005b) but like the resultby LeBlond and Mysak (1978) it was based on the assumption of plane wave solutionsΨ = exp i(kx +mz+ωt) The present result demonstrates that (16) is valid for the generalsolution (15) as well

Under the TA we have fs = 0 and microminus = minusmicro+ so the two characteristics then give thesame ratio namely

Ep

Ek

= N2(ω2 minus f 2)

ω2(N2 minus f 2) + f 2(N2 minus ω2)(TA) (17)

which is a familiar expression (eg LeBlond and Mysak 1978 Eq 875)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 equipar-tition of potential and kinetic energies (ie EpEk = 1) for all frequencies but this is notso in the non-traditional case (16) which at the equator becomes

Ep

Ek

= N2

N2 + 2f 2s

(18)

This value is indicated by the horizontal dotted line in the upper panel of Figure 5Even though (17) holds irrespective of whether F or G is selected it does not hold if

both are involved at the same time This situation occurs in a basin of finite depth (such asthe ocean) where it is not sufficient to include just F because wave reflection from theboundaries (bottom or surface) will result in propagation along the other characteristic and


Figure 5 The ratio of potential and kinetic energies EpEk versus wave frequency ω Here a lowvalue of stratification is chosen N = 4Ω characteristic of the deepest layer of the ocean (cf Fig 1)Results are shown for three different latitudes In each the ratio for the non-traditional expression(16) is shown both for micro+ (thick solid line) and for microminus (thick dotted line) here wave frequenciesmust lie between ωmin and ωmax Also shown is the traditional case (17) thin solid line here thefrequency range is smaller being constrained by the lower bound |f | and upper bound N

hence requires the inclusion of G Near the surface or a horizontal bottom the potentialenergy must vanish (henceEpEk rarr 0) because no vertical excursions can occur there Thismeans (17) is no longer valid With non-traditional terms included the situation becomeseven more complicated because the ratios are then different for F and G being given by(16) and by its counterpart with microminus Either way in this situation the vertical dependence ofEpEk is most easily found in terms of modes as discussed in the next subsection

b Vertical modes

If we consider a vertically confined basin with a horizontal bottom and surface it is moreconvenient to express the solution directly in terms of vertical modes instead of using (15)Thus we write following Saint-Guily (1970)

Ψ = Ψ(z) exp ik(x minus BzC) (19)


substitution of which into (7) gives

d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (20)

Together with the boundary conditions Ψ = 0 at z = 0 minusH (surface bottom) this con-situtes a Sturm-Liouville problem which for given latitude stratification and wave fre-quency yields a discrete set of eigenvalues kn and corresponding vertical modes Ψn Notethat Ψn carries only part of the vertical dependence Ψn the other part being included inexp(minusiknBzC) (The latter disappears under the TA) The other fields can now be expressedas follows

Un = Ψprimen minus ikn

B

CΨn (21)

Vn = if

ωΨprime

n + kn

ωfs

CΨn (22)

Wn = minusiknΨn (23)

Γn = N2 kn

ωΨn (24)

Pn = C

ωkn

Ψprimen + fcΨn (25)

Each has to be multiplied by exp ikn(x minus BzC) + ωt to get the full solution We notethat the expression for pressure (25) contains fc demonstrating that non-traditional effectswould be present even in the case of east-west propagation when fs = 0

If we consider only one mode (modenumber n) then the kinetic and potential energy isgiven by

Ekn = 1

4ρlowast

[1 +

(f

ω

)2] (

Ψprimen

)2 + k2n

[B2 + (ωfs)

2

C2+ 1

]Ψ2

n

Epn = 1

4ρlowastN2

(kn

ω

)2

Ψ2n (26)

These expressions hold generally for any profile of N If we now restrict ourselves to thecase of constant N then we can obtain the expression for the depth-integrated energiesFrom (20) we have after multiplication by Ψ and partial integrationint

dz(Ψprime

n

)2 = k2n

[B2 minus AC

C2

] intdzΨ2

n

After a little algebra the ratio of the depth-averaged potential and kinetic energies for theuni-modal case is found to be

Epn

Ekn

= N2(ω2 minus f 2)2

(ω2 minus f 2)[ω2(N2 minus f 2) + f 2(N2 minus ω2)] + 2ω2f 2s (ω2 + f 2)

(27)


(The bar stands for depth-averaging) Under the TA (fs = 0) this reduces simply to (17)but with non-traditional terms included the expression is different both from (16) and fromits counterpart with microminus This means that neither of the characteristics produces the ratio forthe 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 notmean that the energy ratio is given by (27) in the multi-modal case Under the TA the modalfunction Ψn carries the whole vertical dependence and orthogonality of modes Ψn thenensures that products of modes do not contribute to the expressions for the depth-integratedenergy Hence (17) remains valid in the multi-modal case With fs included however thereis an extra vertical dependence via the exponential factor exp(minusiknBzC) resulting incontributions of products of modes to the depth-integrated energies As a consequence theratio EpEk is no longer given by (27) rather it varies with x

This is shown in Figure 6 Here the first ten modes are superposed resulting in a beam(Fig 6a) The ratio EpEk shown in Figure 6b has a layered structure with values alternat-ing between approximately zero (white) and one (black) This layering reflects the numberof modes involved (there are ten black layers) and would become finer-grained if moremodes were included The layering is however most prominent precisely in regions wherethe total energy is weak and hence is of little significance In the region where the beamoccurs (cf Fig 6a) values are more uniform being low in the descending part of the beamand high in the ascending part This is further illustrated in Figure 6c where the depth-averaged Ep and Ek are used to form the ratio EpEk (solid line) Here the connection withthe direction of the beam is evident the variation in x reflects precisely the transition fromone characteristic to another The alternation between high and low values in Figure 6cas indicated by the solid line does not yet correspond to the values for the characteristicsgiven by (16) and its counterpart with microminus (indicated by the horizontal dotted lines) How-ever numerical evidence demonstrates that the latter values are approached if more modesare taken into account an example is shown by the dashed line in Figure 6c which is basedon 100 modes

Interestingly the expression for the uni-modal case (27) too remains meaningful forif we take not only the vertical average of Ep and Ek but also the horizontal averagewe obtain a ratio Eprime

pEprimek which is always equal to (27) no matter how many modes are

involved No such horizontal averaging would be needed under the TA of course becausein that case the ratio EpEk itself is already horizontally uniform

5 Polarization of the horizontal velocity field

We now consider the horizontal velocity field again applying the methods of character-istics and vertical modes separately Specifically we examine the three key properties ofthe (u v) field which takes the form of an ellipse the ratio of minor to major axis theorientation of the ellipse ie the angle in the horizontal plane between the major axis andthe x-direction and the polarization ie the sense in which the ellipse is traversed in time


Figure 6 A multi-modal example with ten modes included showing a) the location of the beamwhere the energy is concentrated (total energy density Ep +Ek is here normalized arbitrarily) Inb) the ratio EpEk is shown it varies both with x and with z In c) the ratio of the depth-averagedpotential and kinetic energies Ep and Ek which varies with x high (low) values are found wherethe beam follows the direction of the micro+ (microminus) characteristic The levels corresponding to thesecharacteristics are indicated by the two horizontal dotted lines the upper line is based on (16)the lower line on its counterpart with microminus These levels are approached if many more modes aretaken into account an example is shown for 100 modes (dashed line) Parameters are N = 4Ωφ = 45N ω = 14052 times 10minus4 rads (ie semidiurnal M2 tidal frequency) and α = π2 (ienorthward propagation to the right)

We first briefly summarize the familiar results as obtained under the TA (see eg Kundu1976) The most common situation is N gt |f | in which case the ratio of minor to majoraxis is given by |f |ω this follows from (3) with fs = 0 and partparty = 0 Furthermorethe orientation of the major axis is 0 ie it lies in the direction of wave propagation Invery weakly stratified layers however when N lt |f | the ratio of minor to major axis isω|f | (since then N lt ω lt |f | see Fig 4) and the major axis has an angle of 90 withrespect to wave propagation Finally irrespective of whether |f | or N is the largest thepolarization is always clockwise in the Northern Hemisphere and counterclockwise in the


Southern Hemisphere These results are valid both for characteristics and for modes anddo not depend on the number of modes involved As we will see below the situation getsmuch more complicated if one abandons the TA

a Characteristics

With 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 wave propagation which is at anangle α north of east (Fig 2)

Selecting again F we find for u and v

u = minusF prime exp(iωt) v = ωminus1(fsmicro+ minus f )iF prime exp(iωt)where the prime denotes the derivative to the argument of F ie F prime = partFpartξ+ Theseexpressions imply that u and v have a phase difference of 90 (The same argument appliesif G were selected instead of F in which case we have to replace micro+ by microminus) The ratio ofminor to major axis of the u v-hodograph is therefore simply given by

|v||u| = |fsmicroplusmn minus f |ω (28)

(Or if the numerator exceeds the denominator this represents the ratio of major to minoraxis instead)

The polarization is determined by the sign of f minus fsmicroplusmn if positive it is clockwiseif negative counterclockwise Unlike under the TA its sign is no longer determined bythe hemisphere alone the polarization may now become counterclockwise in the NorthernHemisphere The transition between clockwise and counterclockwise polarization occurswhen f = fsmicroplusmn (hence v = 0 the degenerated case of rectilinear polarization) whichimplies using the definition of microplusmn in (9)

ω = |f |N(f 2 + f 2

s

)12 (29)

If we substitute this expression back into f = fsmicroplusmn we find that for this equality to holdwe have to select micro+ in the Northern Hemisphere and microminus in the Southern HemisphereSo for example for inertial waves (ω = |f |) associated with micro+ and propagating in themeridional direction in the Northern Hemisphere the condition (29) implies that the polar-ization becomes rectilinear if N = 2Ω Figure 1 indicates that stratification as weak as thisexists in the deepest layers of the ocean so one would here expect rectilinear polarization ofinertial motions to occur signifying the transition between clockwise and counterclockwisepolarized ellipses of the (u v) field We are however not aware of pertinent observationalresults in which this has been examined

There are some special frequencies that require attention First again the inertial fre-quency ω = |f | in which case microminus = 0 so for this characteristic polarization is alwayscircular and clockwise in the Northern Hemisphere (in contrast to micro+ discussed above)


Second ω = (N2 +f 2s )12 ie A = 0 which implies that microplusmn becomes infinite resulting in

rectilinear polarization in a direction perpendicular to wave propagation Finally ω = N in which case we have microplusmn = (f plusmn ω)fs hence (28) now implies circular polarizationThis holds for any N no matter how large and thus poses a paradox because one would notexpect high-frequency waves to be affected by Coriolis effects This paradox can be resolvedin two ways First we have supposed that waves exist lsquoeternallyrsquo by assuming exp iωt Inreality high-frequency waves are dissipated long before Coriolis effects could have madeany impact (which requires time scales of the order of Ωminus1) Second even if waves wereto live long enough for the horizontal velocity field to become circular this would still beinsignificant because at high frequencies ω asymp N Ω the excursion of the water parcelsis predominantly vertical anyway Still the case ω = N provides an interesting illustrationof the singular nature of the TA as long as the TA is not made the horizontal polarizationis circular even for strong stratification whereas if one makes the TA it becomes nearlyrectilinear at high frequencies

As noted above (28) applies if either F or G is selected However if both F and G areinvolved at the same time then the phase difference between u and v need no longer be90 and the ellipticity can no longer be expressed simply as the ratio between the two Inother words the simplicity of (28) is deceptive because it hinges on the selection of justone of the functions F or G which is possible only in a medium of infinite extension Weelaborate on this point in the following section where we consider a layer of finite verticalextent and use vertical modes

b Vertical modesWe now turn to the properties of the horizontal velocity field in terms of modes Without

loss of generality we can express the (real) horizontal velocity u as

u = ru cos(ωt + φu)

and similarly for v Here amplitudes (ruv gt 0) and phases (φuv) depend on the spatialcoordinates x and z Combining the two we can write

u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)

and cos χ = c1b sin χ = c2b


This implies that the radius of the major axis of the horizontal current field is (a + b)12and the radius of the minor axis (a minus b)12 so the ratio of minor to major axis is(

a minus b

a + b

)12

The major axis is reached when t = tlowast = minusχ(2ω) The orientation of the ellipse iethe angle in the horizontal plane between the major axis and the x-direction is given byarctan(v(tlowast)u(tlowast)) Finally the polarization follows from the sign of sin(φu minus φv) fornegative values the polarization is clockwise for positive values counterclockwise (Thiscan be seen from the fact that u takes its maximum at t = minusφuω and at that instant thesign of partvpartt is equal to that of sin(φu minus φv) whence follows the sense of polarization)

We consider first in Figure 7 the case with the same parameters as in Figure 6 heretoo ten modes are included In Figure 7a we recognize the beam structure from Figure 6a

Figure 7 A multi-modal example with ten modes included showing properties of the horizontalvelocity field (u v) In a) the location of the beam in b) a measure of the ellipticity and in c) theorientation of the major axis in the horizontal plane Parameters are as in Figure 6


now depicted in terms of the magnitude of the major axis Figure 7b demonstrates that theratio of minor to major axis varies spatially values are smaller in the upward than in thedownward beam looking from left to right (Recall that under the TA by contrast the ratiowould be spatially uniform being always equal to |f |ω) Within these beams values areon average close those derived for the characteristics which are given by (28) yielding 091for microminus and 036 for micro+ As one would expect they come closer to the latter values if moremodes are included (not shown) If instead we use fewer modes the reverse happens (Fig8b) With just two modes we can of course no longer speak of a well-defined beam Themost surprising outcome is what happens if we take a single mode only The ratio of minorto major axis then becomes spatially uniform as under the TA moreover it even takes thesame value as under the TA (Fig 9b) Numerically it is found that this happens for anystratification and for any mode n (as long as we restrict ourselves to a single mode) for one

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


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

mode the ratio is always given by the lsquotraditionalrsquo value |f |ω (More illustrations of thisresult are included in Exarchou (2007))

However even for one mode there are in other respects still important departures from thetraditional result Under the TA the first mode would have vanishing u and v halfway downthe water column which clearly is not the case in the non-traditional Figure 9a The reasonsimply lies in the extra (ie second) z dependence in (19) which produces the second termson the right in (21) and (22) Another more surprising departure concerns the orientationof the ellipse Even though the ratio of minor to major axis is in the uni-modal case thesame as under the TA the orientation of the major axis is different and moreover it variesvertically covering all possible angles (Fig 9c) In other words non-traditional effectscreate a tilt of the ellipse If more modes are included as in Figure 8c the orientation varieshorizontally as well but tends to become uniform (in this case zero) within the beam ifmany modes are involved (Fig 7c) This is in agreement with the result for characteristics


presented in Section 5a where we found that each characteristic taken separately gives aphase difference of 90 between u and v which in this case means that the major axis mustbe aligned with u

To summarize we find that for one mode the ratio of minor to major axis is uniform andtakes the traditional value but unlike under the TA the major axis has a tilt that variesvertically For many modes the tilt tends to disappear within the beam but the ratio ofminor and major axis approaches the values corresponding to the characteristics whichunlike under the TA are different for the upward and downward beams

Finally we turn to the polarization For the parameters used above in Figure 7 onlyminor spots of counterclockwise polarization are found around the upward going beams(not shown) To give a clearer example we take a lower latitude the other parametersare unchanged so this still describes a semidiurnal internal tide beam propagating in themeridional direction through a weakly stratified layer In Figure 10b we see that the upwardgoing beam (looking from left to right) has counterclockwise polarization throughout (Asa matter of fact f minus fsmicro+ is now negative so one would indeed expect counterclockwisepolarization in the beam associated with the micro+ characteristic however this criterion doesnot always work at higher latitudes one may find spots of counterclockwise polarizationeven though f minus fsmicroplusmn are both positive) This result demonstrates clearly that given theright circumstances internal waves may have counterclockwise polarization in the NorthernHemisphere Lowering the number of modes gives qualitatively the same result as long asthere is more than one however numerical evidence suggests that for a single mode thepolarization is always as under the TA (ie clockwise in the Northern Hemisphere)

6 Propagation through layers of different stratification

We now consider the propagation through layers of different stratification N Specificallywe adopt a three-layer system in which the upper and lower layer are of semi-infinite extentwhile the middle layer has a thickness h The upper and lower layer have N(z) = N1 andthe middle layer N(z) = N2 where N12 are constants (Fig 11) We want to determine howmuch of the incoming energy (from below say) is transmitted through the middle layerand how this depends on the parameters of the system and on making the TA or not

To solve this problem we return to (7) in which A now depends on z because N varieswith z However within each layer A is constant and for later reference we define theconstants

A12 = N212 minus ω2 + f 2

s

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

d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)

Within each layer the part in square brackets is constant so the solution is either sinusoidalor exponential depending on the sign of B2 minusAC We assume that waves are sinusoidal in


Figure 10 A multi-modal example with 10 modes included showing properties of the horizontalvelocity field (u v) In (a) the location of the beam in (b) the polarization where positive (negative)values mean that the polarization is counterclockwise (clockwise) Parameters are as in Figure 6except for the latitude which is now φ = 15N

Figure 11 A three layer system consisting of semi-infinite upper and lower layers with N(z) = N1and a middle layer of thickness h and stratification N(z) = N2 Waves enter the system from below


the upper and lower layers ie wave frequencies must be such that B2 minus A1C gt 0 For themiddle layer we assume for the moment that the solution is exponential ie B2 minusA2C lt 0The solution of (35) can thus be written

Ψ =⎧⎨⎩

c1eiqz + c2e

minusiqz (z lt minush2 lower layer)c3e

κz + c4eminusκz (minush2 lt z lt h2 middle layer)

c5eiqz + c6e

minusiqz (z gt h2 upper layer)(36)

where

q = k∆

121

C κ = k

(minus∆2)12

C

with q and κ both real and ∆12 = B2 minus A12CAt the levels z = plusmnh we have to require continuity of Ψ and of its derivative Ψprime

thus guaranteeing continuity of the vertical velocity and pressure given by (23) and (25)respectively This allows us to express the coefficients c12 in terms of c56 Assuming wavesto enter from below we may take c6 = 0 The problem is now mathematically identicalto that of a rectangular potential barrier in quantum mechanics (Merzbacher 1998 sect62)which yields the following expression for the transmission coefficient T = |c5|2|c1|2

T = 1

cosh2 κh minus (∆1+∆2)2

4∆1∆2sinh2 κh

= 1

cosh2[kh(minus∆2)12C] minus (∆1+∆2)2

4∆1∆2sinh2[kh(minus∆2)12C]

(37)

An expression equivalent to (37) was used before by Sutherland and Yewchuk (2004) butthere ∆12 were defined differently because Coriolis effects were ignored altogether Herewe include them and examine the effect of the TA

So far we have assumed that Ψ is exponential in the middle layer (ie ∆2 lt 0 and κ real)Eq (37) applies however also when ∆2 gt 0 in which case we can write

T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)

Thus both regimes of the middle layer can be considered For given latitude and strati-fication the transmission coefficient T in (37) and (38) depends on two parameters onwave frequency ω (via ∆12 and C) and on kh the thickness of the middle layer measuredagainst the horizontal wavelength

In the figures that follow we plot T as a function of these two parameters In each figurewe take N1 = 4Ω φ = 45N and α = π2 (propagation in the meridional direction) OnlyN2 the stratification of the middle layer is varied we take N2 = 0 (ie a convective layer)in Figure 12 N2 = 2Ω in Figure 13 and N2 = 6Ω in Figure 14 In each the non-traditionalresult is shown in the upper panel the traditional result in the lower panel


Figure 12 The transmission coefficient T as a function of scaled wave frequency ωf and of scaledmiddle-layer thickness kh Latitude is φ = 45N and the outer layers have stratification N1 = 4ΩHere the middle layer is convective N2 = 0 The upper panel shows the non-traditional resultthe horizontal axis covers exactly the non-traditional frequency range (ωmin ωmax) that is basedon N1 (outer layers) The arrow indicates the transition from oscillatory to evanescent behaviourin the middle layer Under the TA the frequency range is of course smaller and runs from f to N1as indicated by the vertical dashed lines in the lower panel here waves are always evanescent inthe middle layer

With a convective middle layer (Fig 12) the contrast is starkest Under the TA wavesare evanescent in the middle layer at all frequencies and as a result little energy passesthrough it except for very thin layers (kh 13 1) Without the TA on the other hand energyat low frequencies ie near f passes the middle layer virtually unhindered regardless ofthe thickness of the middle layer In Figure 12a the transition between wave frequencies forwhich ∆2 gt 0 (propagation in the middle layer) on the left and those for which ∆2 lt 0(evanescence) on the right is clearly seen (and indicated by an arrow) Even when wavescan propagate through the middle layer not all the incoming energy passes through itbecause internal reflections occur at the levels separating the layers where N changes from


Figure 13 As in Figure 12 but now with weak stratification in the middle layer N2 = 2Ω

one value to another The intensity of the internal reflections depends on the thickness of themiddle layer as Figure 12a shows We notice that in the ocean changes from a stratified toa convective layer can be quite abrupt (as illustrated for example by van Haren and Millot(2004)) which suggests that the schematic configuration used here is not unreasonable

With a stronger stratification in the middle layer (Fig 13) there is some wave propa-gation through the middle layer at low frequencies under the TA but without the TA thetransmission is higher at all frequencies especially so at near-inertial ones This is still truewhen the stratification of the middle layer exceeds that of the outer layers (Fig 14) Inthis case waves can propagate through the middle layer at all frequencies except the verylowest in Figure 14a Still there is no full transmission at higher frequencies because ofinternal reflections as noted above

The configuration of Figure 11 in which the outer layers have the same stratificationis not always appropriate for the ocean More generally if one replaces the upper layer byone with stratification N3 (instead of N1) and accordingly the solution in the upper layerin (36) by c5 exp(irz) (c6 being again taken to be zero) with r = k∆

123 C then one finds


Figure 14 As in Figure 12 but with a middle-layer stratification exceeding that of the outer layersN2 = 6Ω

c5

c1= 2κq exp[minusi(q + r)h2]

κ(q + r) cosh(κh) + i(κ2 minus qr) sinh(κh)

As before the transmission coefficient is given by T = |c5c1|2 We consider an examplethat is relevant for the ocean We take N1 = 0 representing a (convectively) mixed layerin our set-up this is the layer from which the waves originate We take the middle layer tobe stratified with N2 = 5 times 10minus3 rads and the upper layer to be more weakly stratifiedN3 = 2 times 10minus3 rads In the present set-up waves enter from the lower layer (N1) but theresult would of course be the same if we reverse the vertical placing the mixed layer atthe top in which case we can interpret the N2 layer as a schematic representation of thepermanent pycnocline and the N3 layer as the weakly stratified deep ocean The questionis then what happens to near-inertial waves generated in the top layer Under the TA anyinternal waves propagating in the mixed layer (where N = 0) cannot also propagate in thelower layers where N gt |f | because the corresponding frequency regimes are disjunct(Fig 4) So there will be no transmission down to the other layers Without the TA however


Figure 15 The transmission coefficient T calculated without making the TA as a function of scaledwave frequency ωf and of scaled middle-layer thickness kh Here the three layers are all dif-ferently stratified Waves enter from a mixed layer with N1 = 0 then encounter a stratified middlelayer with N2 = 5 times 10minus3 rads (lsquopycnoclinersquo) and finally the other outer layer which is weaklystratified N3 = 2 times 10minus3 rads (lsquodeep oceanrsquo) Only the near-inertial band is shown for lower orhigher frequencies the transmission coefficient is zero or nearly so The arrow shows the lowerbound of the range of allowable frequencies in the middle layer Latitude is φ = 45N

there is a narrow band around |f | for which internal waves can propagate throughout thevertical as shown in Figure 15

7 Conclusion

It was argued by Garrett (2001) that the near-inertial band stands out in the internal-wavespectrum Not only is it the most energetic part of the spectrum but also the propagation isdifferent near-inertial waves undergo as it were a shift within the band when they propagatein the meridional direction because the meaning of ldquoinertialrdquo changes with latitude Here wehave analysed another aspect of meridional propagation namely the passage through layersof different stratification (Section 6) Here too the near-inertial band is special because it isthe only part of the spectrum that can pass layers of any stratification This means also thatmeridional propagation is favored because no such nontraditional effects occur for zonalpropagation

This is especially remarkable in the case of convective layers (with N zero) beneath orabove a layer of significant stratification Under the TA no frequencies would exist for whichwaves can propagate in both layers but with non-traditional effects included this becomespossible for near-inertial waves Observational evidence by van Haren and Millot (2004)confirms the latter in the Mediterranean Sea near-inertial were observed in both kinds of


layers at the same time They also found that the polarization of the horizontal velocitywas very different in the convective layer instead of being circular it was ellipse-likeAs noted before (Gerkema et al 2008) this is in accordance with non-traditional theorybecause the expression (28) in terms of the micro+ characteristic (which is associated withvertical energy propagation the other one microminus giving horizontal propagation at the inertialfrequency) predicts such a departure from circular polarization However the analysis ofSection 5 shows that the situation is actually more complicated if we take into accountthe vertical confinement of the basin The result is then found to depend strongly on thenumber of modes for one mode polarization would be circular at the inertial frequency itwould become an ellipse only if more modes were included but then still varies stronglyspatially Yet the non-traditional theory is confirmed because under the TA the polarizationwould always be circular at the inertial frequency We also note the complication of thetilt of the ellipse engendered by non-traditional effects which implies that the direction ofwave propagation can no longer be inferred from the orientation of the major axis Also weshowed that the polarization can be counterclockwise in the Northern Hemisphere (Fig 10)

For all the aspects considered here there are qualitative differences depending on whetherone uses characteristics or a number of vertical modes or just one mode So far as the char-acteristics are concerned the results derived above hold only if the entire wavefield dependson only one of the characteristic coordinates This situation is unlikely to be applicable tothe ocean because reflections from the bottom or surface brings the other characteristicinto play as well Regarding the vertical modes near a wavesource a number of modes areexpected to be important and so the multi-modal results obtained above should be relevantthere On the other hand if the higher modes have been dissipated so that only one modeprevails (in the ldquofar fieldrdquo) the unimodal results apply As pointed out above there arequalitative changes if one goes from the multi-modal to the unimodal case For examplethe expression (27) for the vertically averaged energies applies only for a single modehorizontal variations occur as soon as one introduces more modes (Fig 6c) Regarding thepolarization of the horizontal velocity field the ratio of minor to major axis as well as thesense of rotation are lsquotraditionalrsquo in the unimodal case but become different as soon asmore modes are present

A final remark concerns our findings on the ratio of potential and kinetic energies Forweak stratification there is a significant departure from the ratio as obtained under the TA(see eg Fig 5) Apart from being of fundamental interest this modification should alsobe relevant to applications in which the expression for internal-wave energy features Acase in point is the parameterization used to calculate vertical eddy diffusivity from CTDand LADCP profiles (eg Kunze et al 2006) This parameterization is likely to requiremodification in weakly stratified layers as well as in the equatorial region

Acknowledgments We thank Victor Shrira Hans van Haren and Leo Maas for inspiring discussionson the subject matter of this paper


APPENDIX

Ratio of potential and kinetic energies

For plane waves LeBlond and Mysak (1978) derived an expression for the ratio ofpotential and kinetic energies with non-traditional terms included namely

Ep

Ek

= N2 cos2 θ

ω2 + 4Ω2 cos2 αprime (39)

(their equation 865) Here θ denotes the angle of the wavevector k with the horizontal planeand αprime the angle between the wavevector and the rotation vector Ω We here demonstratethat this expression is equivalent to our (16)

We start with the identity cot θ = minusmicroplusmn (the dispersion relation in implicit form) whichwas derived by Gerkema and Shrira (2005a) Hence cos θ = minusmicroplusmn(1 + micro2plusmn)12 Using theinner product we can write k middotΩ = |k|Ω cos αprime With k = |k|(cos θ 0 sin θ) and Ω = f2with f defined by (1) we then find

cos αprime = cos θ sin α cos φ + sin θ sin φ = minusmicroplusmnfs + f

2Ω(1 + microplusmn)12

Hence (39) can also be written as

Ep

Ek

= N2micro2plusmn

ω2(1 + micro2

plusmn) + (fsmicroplusmn minus f )2

(40)

which is precisely (16) and its counterpart with microminus

REFERENCESExarchou E 2007 Propagation of internal waves through convective layers in the ocean MSc thesis

Utrecht UniversityGarrett C 2001 What is the ldquonear-inertialrdquo band and why is it different from the rest of the internal

wave spectrum J Phys Oceanogr 31 962ndash971Gerkema T and V I Shrira 2005a Near-inertial waves in the ocean beyond the lsquotraditional approx-

imationrsquo J Fluid Mech 529 195ndash219mdashmdashmdash 2005b Near-inertial waves on the ldquonon-traditionalrdquo β plane J Geophys Res 110 C01003

doi1010292004JC002519Gerkema T J T F Zimmerman L R M Maas and H van Haren 2008 Geophysical and

astrophysical fluid dynamics beyond the traditional approximation Rev Geophys 46 RG2004doi1010292006RG000220

Kundu P K 1976 An analysis of inertial oscillations observed near Oregan coast J Phys Oceanogr6 879ndash893

Kunze E E Firing J M Hummon T K Chereskin and A M Thurnherr 2006 Global abyssalmixing inferred from lowered ADCP shear and CTD strain profiles J Phys Oceanogr 36 1553ndash1576

LeBlond P H and L A Mysak 1978 Waves in the Ocean Elsevier 602 ppMerzbacher E 1998 Quantum Mechanics 3rd ed Wiley 656 ppSaint-Guily B 1970 On internal waves effects of the horizontal component of the Earthrsquos rotation

and of a uniform current Dtsch Hydrogr Z 23 16ndash23


Sutherland B R and K Yewchuk 2004 Internal wave tunneling J Fluid Mech 511 125ndash134van Haren H 2006 Asymmetric vertical internal wave propagation Geophys Res Lett 33 L06618

doi1010292005GL025499van Haren H and C Millot 2004 Rectilinear and circular inertial motions in the Western Mediter-

ranean sea Deep-Sea Res I 51 1441ndash1455

Received 1 July 2008 revised 5 November 2008


Strong stratification tends to suppress lsquonon-traditionalrsquo effects This can be understoodintuitively from the fact that the Coriolis terms proportional to the cosine of latitude (iethe non-traditional terms) always involve the vertical direction the associated force either isitself vertical or is induced by vertical velocity Since stable vertical stratification in densitytends to reduce vertical movements the effect of the non-traditional Coriolis componentstoo is reduced Generally non-traditional effects can be neglected if stratification is strongin the sense that N 2Ω where N is the buoyancy frequency and Ω the Earthrsquos angularvelocity As a matter of fact this condition is not satisfied in the deepest layers of the oceanas is illustrated in Figure 1

It has been known for a long time that the range of allowable internal-wave frequenciesis extended by including the non-traditional Coriolis terms (LeBlond and Mysak 1978)Within this range three intervals can be distinguished in which internal waves behave dif-ferently with regard to the horizontal or vertical opposition of energy and phase propagation(van Haren 2006) This classification is completed here (Section 3) The starting point isthe recognition that two stratification regimes need to be distinguished according to whichof the two |f | or (N2 + f 2

s )12 is the largest (Here f = 2Ω sin φ with latitude φ andfs = 2Ω cos φ sin α where α is the angle of wave propagation in the horizontal geograph-ical plane north of east) In each regime the range of allowable frequencies includes a

Figure 1 The relative strength of stratification N2Ω derived from temperature and salinity profilesin the Pacific Ocean for a south-north section near 179E (WOCE section P14 from the Fiji Islandsto the Bering Sea) Here N is calculated using its definition N2 = g2(partρ0partp minus cminus2

s ) where ρ0is in-situ density p is pressure and cs the speeed of sound Note the logarithmic gray scaling Inthe deepest layers the condition N2Ω 1 is not satisfied Here non-traditional effects can beexpected to significantly affect internal-wave dynamics the same applies to the upper mixed layerAfter Gerkema et al (2008)














ux + vy + wz = 0 (5)

bt + N2w = 0 (6)



ψ = Ψ exp(iωt)



where





with

ωminmax = 1radic2

([N2 + f 2 + f 2

s

] ∓ [N2 + f 2 + f 2

s

]2 minus (2f N)212)12 (8)




A (9)





ω = |f | ω = (N2 + f 2

s

)12


inequalities


s

)12lt max

|f | (N2 + f 2s

)12lt ωmax (10)
















(N2 + f 2s )12 lt |f | (N2 + f 2















V = i


Γ = minusiN2

ωΨx (12)







Ek = 1

2ρlowast〈(u)2 + (v)2 + (w)2〉

= 1


Ep = 1


= 1






a Characteristics


Ψ = F(ξ+) + G(ξminus) (15)


Ep

Ek

= N2micro2+

ω2(1 + micro2


(16)




Ep

Ek

= N2(ω2 minus f 2)




Ep

Ek

= N2

N2 + 2f 2s

(18)






b Vertical modes





d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (20)



B

CΨn (21)

Vn = if

ωΨprime

n + kn

ωfs

CΨn (22)


Γn = N2 kn

ωΨn (24)

Pn = C

ωkn




Ekn = 1

4ρlowast

[1 +

(f

ω

)2] (

Ψprimen

)2 + k2n

[B2 + (ωfs)

2

C2+ 1

]Ψ2

n

Epn = 1

4ρlowastN2

(kn

ω

)2

Ψ2n (26)


dz(Ψprime

n

)2 = k2n

[B2 minus AC

C2

] intdzΨ2

n


Epn

Ekn



(27)















a Characteristics







ω = |f |N(f 2 + f 2

s

)12 (29)











u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)




a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)






























ux + vy + wz = 0 (5)

bt + N2w = 0 (6)



ψ = Ψ exp(iωt)



where





with

ωminmax = 1radic2

([N2 + f 2 + f 2

s

] ∓ [N2 + f 2 + f 2

s

]2 minus (2f N)212)12 (8)




A (9)





ω = |f | ω = (N2 + f 2

s

)12


inequalities


s

)12lt max

|f | (N2 + f 2s

)12lt ωmax (10)
















(N2 + f 2s )12 lt |f | (N2 + f 2















V = i


Γ = minusiN2

ωΨx (12)







Ek = 1

2ρlowast〈(u)2 + (v)2 + (w)2〉

= 1


Ep = 1


= 1






a Characteristics


Ψ = F(ξ+) + G(ξminus) (15)


Ep

Ek

= N2micro2+

ω2(1 + micro2


(16)




Ep

Ek

= N2(ω2 minus f 2)




Ep

Ek

= N2

N2 + 2f 2s

(18)






b Vertical modes





d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (20)



B

CΨn (21)

Vn = if

ωΨprime

n + kn

ωfs

CΨn (22)


Γn = N2 kn

ωΨn (24)

Pn = C

ωkn




Ekn = 1

4ρlowast

[1 +

(f

ω

)2] (

Ψprimen

)2 + k2n

[B2 + (ωfs)

2

C2+ 1

]Ψ2

n

Epn = 1

4ρlowastN2

(kn

ω

)2

Ψ2n (26)


dz(Ψprime

n

)2 = k2n

[B2 minus AC

C2

] intdzΨ2

n


Epn

Ekn



(27)















a Characteristics







ω = |f |N(f 2 + f 2

s

)12 (29)











u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)




a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)























ux + vy + wz = 0 (5)

bt + N2w = 0 (6)



ψ = Ψ exp(iωt)



where





with

ωminmax = 1radic2

([N2 + f 2 + f 2

s

] ∓ [N2 + f 2 + f 2

s

]2 minus (2f N)212)12 (8)




A (9)





ω = |f | ω = (N2 + f 2

s

)12


inequalities


s

)12lt max

|f | (N2 + f 2s

)12lt ωmax (10)
















(N2 + f 2s )12 lt |f | (N2 + f 2















V = i


Γ = minusiN2

ωΨx (12)







Ek = 1

2ρlowast〈(u)2 + (v)2 + (w)2〉

= 1


Ep = 1


= 1






a Characteristics


Ψ = F(ξ+) + G(ξminus) (15)


Ep

Ek

= N2micro2+

ω2(1 + micro2


(16)




Ep

Ek

= N2(ω2 minus f 2)




Ep

Ek

= N2

N2 + 2f 2s

(18)






b Vertical modes





d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (20)



B

CΨn (21)

Vn = if

ωΨprime

n + kn

ωfs

CΨn (22)


Γn = N2 kn

ωΨn (24)

Pn = C

ωkn




Ekn = 1

4ρlowast

[1 +

(f

ω

)2] (

Ψprimen

)2 + k2n

[B2 + (ωfs)

2

C2+ 1

]Ψ2

n

Epn = 1

4ρlowastN2

(kn

ω

)2

Ψ2n (26)


dz(Ψprime

n

)2 = k2n

[B2 minus AC

C2

] intdzΨ2

n


Epn

Ekn



(27)















a Characteristics







ω = |f |N(f 2 + f 2

s

)12 (29)











u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)




a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)




















A (9)





ω = |f | ω = (N2 + f 2

s

)12


inequalities


s

)12lt max

|f | (N2 + f 2s

)12lt ωmax (10)
















(N2 + f 2s )12 lt |f | (N2 + f 2















V = i


Γ = minusiN2

ωΨx (12)







Ek = 1

2ρlowast〈(u)2 + (v)2 + (w)2〉

= 1


Ep = 1


= 1






a Characteristics


Ψ = F(ξ+) + G(ξminus) (15)


Ep

Ek

= N2micro2+

ω2(1 + micro2


(16)




Ep

Ek

= N2(ω2 minus f 2)




Ep

Ek

= N2

N2 + 2f 2s

(18)






b Vertical modes





d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (20)



B

CΨn (21)

Vn = if

ωΨprime

n + kn

ωfs

CΨn (22)


Γn = N2 kn

ωΨn (24)

Pn = C

ωkn




Ekn = 1

4ρlowast

[1 +

(f

ω

)2] (

Ψprimen

)2 + k2n

[B2 + (ωfs)

2

C2+ 1

]Ψ2

n

Epn = 1

4ρlowastN2

(kn

ω

)2

Ψ2n (26)


dz(Ψprime

n

)2 = k2n

[B2 minus AC

C2

] intdzΨ2

n


Epn

Ekn



(27)















a Characteristics







ω = |f |N(f 2 + f 2

s

)12 (29)











u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)




a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)

























(N2 + f 2s )12 lt |f | (N2 + f 2















V = i


Γ = minusiN2

ωΨx (12)







Ek = 1

2ρlowast〈(u)2 + (v)2 + (w)2〉

= 1


Ep = 1


= 1






a Characteristics


Ψ = F(ξ+) + G(ξminus) (15)


Ep

Ek

= N2micro2+

ω2(1 + micro2


(16)




Ep

Ek

= N2(ω2 minus f 2)




Ep

Ek

= N2

N2 + 2f 2s

(18)






b Vertical modes





d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (20)



B

CΨn (21)

Vn = if

ωΨprime

n + kn

ωfs

CΨn (22)


Γn = N2 kn

ωΨn (24)

Pn = C

ωkn




Ekn = 1

4ρlowast

[1 +

(f

ω

)2] (

Ψprimen

)2 + k2n

[B2 + (ωfs)

2

C2+ 1

]Ψ2

n

Epn = 1

4ρlowastN2

(kn

ω

)2

Ψ2n (26)


dz(Ψprime

n

)2 = k2n

[B2 minus AC

C2

] intdzΨ2

n


Epn

Ekn



(27)















a Characteristics







ω = |f |N(f 2 + f 2

s

)12 (29)











u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)




a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)




























V = i


Γ = minusiN2

ωΨx (12)







Ek = 1

2ρlowast〈(u)2 + (v)2 + (w)2〉

= 1


Ep = 1


= 1






a Characteristics


Ψ = F(ξ+) + G(ξminus) (15)


Ep

Ek

= N2micro2+

ω2(1 + micro2


(16)




Ep

Ek

= N2(ω2 minus f 2)




Ep

Ek

= N2

N2 + 2f 2s

(18)






b Vertical modes





d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (20)



B

CΨn (21)

Vn = if

ωΨprime

n + kn

ωfs

CΨn (22)


Γn = N2 kn

ωΨn (24)

Pn = C

ωkn




Ekn = 1

4ρlowast

[1 +

(f

ω

)2] (

Ψprimen

)2 + k2n

[B2 + (ωfs)

2

C2+ 1

]Ψ2

n

Epn = 1

4ρlowastN2

(kn

ω

)2

Ψ2n (26)


dz(Ψprime

n

)2 = k2n

[B2 minus AC

C2

] intdzΨ2

n


Epn

Ekn



(27)















a Characteristics







ω = |f |N(f 2 + f 2

s

)12 (29)











u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)




a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)






















Ek = 1

2ρlowast〈(u)2 + (v)2 + (w)2〉

= 1


Ep = 1


= 1






a Characteristics


Ψ = F(ξ+) + G(ξminus) (15)


Ep

Ek

= N2micro2+

ω2(1 + micro2


(16)




Ep

Ek

= N2(ω2 minus f 2)




Ep

Ek

= N2

N2 + 2f 2s

(18)






b Vertical modes





d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (20)



B

CΨn (21)

Vn = if

ωΨprime

n + kn

ωfs

CΨn (22)


Γn = N2 kn

ωΨn (24)

Pn = C

ωkn




Ekn = 1

4ρlowast

[1 +

(f

ω

)2] (

Ψprimen

)2 + k2n

[B2 + (ωfs)

2

C2+ 1

]Ψ2

n

Epn = 1

4ρlowastN2

(kn

ω

)2

Ψ2n (26)


dz(Ψprime

n

)2 = k2n

[B2 minus AC

C2

] intdzΨ2

n


Epn

Ekn



(27)















a Characteristics







ω = |f |N(f 2 + f 2

s

)12 (29)











u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)




a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)



















a Characteristics


Ψ = F(ξ+) + G(ξminus) (15)


Ep

Ek

= N2micro2+

ω2(1 + micro2


(16)




Ep

Ek

= N2(ω2 minus f 2)




Ep

Ek

= N2

N2 + 2f 2s

(18)






b Vertical modes





d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (20)



B

CΨn (21)

Vn = if

ωΨprime

n + kn

ωfs

CΨn (22)


Γn = N2 kn

ωΨn (24)

Pn = C

ωkn




Ekn = 1

4ρlowast

[1 +

(f

ω

)2] (

Ψprimen

)2 + k2n

[B2 + (ωfs)

2

C2+ 1

]Ψ2

n

Epn = 1

4ρlowastN2

(kn

ω

)2

Ψ2n (26)


dz(Ψprime

n

)2 = k2n

[B2 minus AC

C2

] intdzΨ2

n


Epn

Ekn



(27)















a Characteristics







ω = |f |N(f 2 + f 2

s

)12 (29)











u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)




a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)




















b Vertical modes





d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (20)



B

CΨn (21)

Vn = if

ωΨprime

n + kn

ωfs

CΨn (22)


Γn = N2 kn

ωΨn (24)

Pn = C

ωkn




Ekn = 1

4ρlowast

[1 +

(f

ω

)2] (

Ψprimen

)2 + k2n

[B2 + (ωfs)

2

C2+ 1

]Ψ2

n

Epn = 1

4ρlowastN2

(kn

ω

)2

Ψ2n (26)


dz(Ψprime

n

)2 = k2n

[B2 minus AC

C2

] intdzΨ2

n


Epn

Ekn



(27)















a Characteristics







ω = |f |N(f 2 + f 2

s

)12 (29)











u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)




a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)



















d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (20)



B

CΨn (21)

Vn = if

ωΨprime

n + kn

ωfs

CΨn (22)


Γn = N2 kn

ωΨn (24)

Pn = C

ωkn




Ekn = 1

4ρlowast

[1 +

(f

ω

)2] (

Ψprimen

)2 + k2n

[B2 + (ωfs)

2

C2+ 1

]Ψ2

n

Epn = 1

4ρlowastN2

(kn

ω

)2

Ψ2n (26)


dz(Ψprime

n

)2 = k2n

[B2 minus AC

C2

] intdzΨ2

n


Epn

Ekn



(27)















a Characteristics







ω = |f |N(f 2 + f 2

s

)12 (29)











u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)




a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)































a Characteristics







ω = |f |N(f 2 + f 2

s

)12 (29)











u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)




a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)






















a Characteristics







ω = |f |N(f 2 + f 2

s

)12 (29)











u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)




a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)



















a Characteristics







ω = |f |N(f 2 + f 2

s

)12 (29)











u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)




a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)

























u2 + v2 = a + b cos(2ωt + χ) (30)

with

a = 1

2

(r2u + r2

v

)(31)

b = (c2

1 + c22

)12(32)

c1 = 1

2

[r2u cos(2φu) + r2

v cos(2φv)]

(33)

c2 = 1

2

[r2u sin(2φu) + r2

v sin(2φv)]

(34)




a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)



















a minus b

a + b

)12


















A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)































A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)




























A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)
























A12 = N212 minus ω2 + f 2

s


d2Ψ

dz2+ k2

[B2 minus AC

C2

]Ψ = 0 (35)







Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)






















Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)



















Ψ =⎧⎨⎩

c1eiqz + c2e



c5eiqz + c6e


where

q = k∆

121

C κ = k

(minus∆2)12

C



T = 1


4∆1∆2sinh2 κh

= 1



(37)



T = 1

cos2[kh∆

122 C

] + (∆1+∆2)2

4∆1∆2sin2 [

kh∆122 C

] (38)














c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)




























c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)

























c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)



















c5







7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)




















7 Conclusion









APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)























APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)


















APPENDIX



Ep

Ek

= N2 cos2 θ







Ep

Ek

= N2micro2plusmn

ω2(1 + micro2


(40)






















Internal-wave properties in weakly stratiÞed layers · vertically bounded domain, using vertical modes. Here, one needs to further distinguish between using a single mode, or more

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