Steady and unsteady nonlinear internal waves incident on an interface

Steady and unsteady nonlinear internalwaves incident on an interface

Roger GrimshawDepartment of Mathematical Sciences,

Loughborough University, U.K.

John McHughDepartment of Mechanical Engineering

University of New Hampshire, U.S.A.

July 4, 2012

Abstract

Steady nonlinear internal waves are commonly described by the Dubreil-Jacotin-Long equation. This equation contains unknown functions of the stream function,representing the density and vorticity fields. Often these are determined by up-stream conditions where the flow is assumed to be known. But for the case whenthe waves are periodic in the horizontal direction, these functions need to be deter-mined instead by consideration of the source of the waves, and in particular by thewave-induced mean flow. Here we show that this situation is particularly importantfor waves incident and reflected from an interface, representing a sharp change in thebackground density stratification, such as that at the tropopause. The combinationof the incident and reflected wave-induced mean flows generates a sharp shear nearthe interface.

1 Introduction

The Earth’s atmosphere has a sudden increase in static stability, measured by the buoy-ancy frequency N(z), with increasing altitude z at the tropopause, and again at themesopause, with a sudden decrease in stability at the stratopause. These sudden changesin the static stability are potential barriers for vertically propagating internal waves, withimportant implications for weather, climate, and aviation. An idealized but useful modelof the dynamics at these altitudes is a two-layer flow with constant buoyancy frequency ineach layer. For instance, this model represents the tropopause as an interface where thedensity gradient is discontinuous while the density remains continuous. This interfacialmodel has been used extensively in the treatment of nonlinear steady mountain waves,

Fully nonlinear steady two-dimensional flows are described by the Dubreil-Jacotin-Long (DJL) equation [1, 2] which has been extensively used in the literature for the studyof mountain waves. For instance Ikawa [3] used the two-layer model in the DJL equation

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for weakly nonlinear mountain waves, while Durran [4] using the same model obtainedfully nonlinear mountain waves with an iterative technique. An important feature of theDJL equation is that in general it contains an arbitrary function of the stream function,which for mountain waves is usually determined by assuming that all streamlines emanatefrom upstream, where the flow reduces too a known basic state. Recently McHugh [5]considered steady monochromatic periodic waves using the DJL equation for a two-layerflow, and found that when a wave is incident on the interface, the combination of theincident and reflected wave generated a mean flow, proportional to the square of the waveamplitude. Because, in this inviscid theory, this mean flow is discontinuous across theinterface, there is an implication that this will lead to shear instability and turbulence nearthe interface. However, for horizontally periodic waves, there are no upstream conditionsthat can be used to determine the arbitrary functions in the DJL equation, and insteadMcHugh [5] determined them using the basic state when the waves are absent. In theBoussinesq approximation, the DJL equation is then a linear equation on each layer,since the buoyancy frequency is constant in each layer, and all nonlinearity resides in theinterface conditions, which for steady flow are that the interface is a streamline, acrosswhich the pressure is continuous. The main purpose of this paper is to restore wavetransience, and so reconsider this problem as the long-time steady outcome of a well-posed initial value problem. It will transpire that then the arbitrary functions in the DJLequation can be uniquely determined, but are different from those used in [5].

Unsteady vertically propagating internal waves in a single layer have been consideredby Grimshaw [6], Shrira [7], Voronovich, [8], Sutherland [9] and Tabaei and Akylas [10] intheoretical studies and Sutherland [11] performed direct numerical simulations. From theperspective of this paper the most important finding is that a vertically propagating wavepacket generates a mean flow that is localised to the wave packet. Recently, the unsteadydevelopment of internal waves in a two-layer inviscid flow was considered in numericalsimulations by McHugh [12] for horizontally periodic monochromatic waves, with a mod-ulated amplitude in the vertical, and by McHugh and Sharman [13] for mountain waves.These results show that the wave packet creates a mean flow that is strongest at theinterface. This generated mean flow may even be strong enough to form a critical level,causing further incident waves to break. Further ongoing work by McHugh [14] treats theunsteady development of the waves assuming the wave amplitude is slowly varying in thevertical. The unsteady theory shows that the combination of incident waves and reflectedwaves in the lower layer results in a brief period where both are creating a mean flowthat is additive, creating a jet-like feature that is similar to the results from the numericalsimulations. Furthermore, the mean flow is found to be discontinuous at the interface.

Here we reconsider the steady two-layer problem (as in [5]), but include the unsteadydevelopment of the waves from an initial value problem. In section 2 we formulate theproblem in the Boussinesq approximation. Then in section 3 we briefly recall the linearizedsolution, describe how the mean flow can be determined, and extend the solution to secondorder in wave amplitude. Then in section 4 we revisit the steady formulation using the DJLequation, but with a new determination of the arbitrary functions using our previouslydetermined mean flow results. We conclude in section 5.

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2 Formulation

We consider the two-dimensional flow of an inviscid incompressible fluid. The full equa-tions for the perturbation variables are, in standard notation,

ρ0ut + px = F1 = −ρ0(uux + wuz)− ρ(ut + uux + wuz) , (1)

ρ0wt + pz + gρ = G1 = −ρ0(uwx + wwz)− ρ(wt + uwx + wwz) , (2)

ux + wz = 0 , (3)

ζt − w = H1 = −uζx − wζz , (4)

gρt − wρ0N2(z) = J1 = −ugρx − wgρz , (5)

Here p0(z), ρ0(z) are the background pressure and density, p0z(z) = −gρ0(z), and ρ0N2 =

−gρ0z. In the sequel N2 will have a discontinuity across z = η, where the boundaryconditions are

ζ = η , at z = η , (6)

[p+ p0(z)]+− = 0 , at z = η . (7)

The density equation (5) can be solved by ρ0(z) + ρ = ρ0(z − ζ), so that to leading orderin wave amplitude,

ρ = −ρ0zζ +ρ0zzζ

2

2+ · · · . (8)

The omitted terms are O(ζ3). The boundary conditions (6, 7) can be expanded so that

[p− gρ0η + pzη +ρ0N

2η2

2+ · · · ]+− = 0 , ζ + ηζz + · · · = η , at z = 0 ,

and then combined to give

[p− gρ0ζ]+− = K1 = [gρ0ζζz +ρ0N

2ζ2

2+ · · · ]+− at z = 0 . (9)

Thus the variables ρ, η are formally eliminated. Further, to leading order in wave ampli-tude, it is readily shown ζz is continuous across the interface, and so (9) reduces to

[p− gρ0ζ]+− =ρ0ζ

2

2[N2]+− + · · · , at z = 0 . (10)

In a Boussinesq fluid, we set ρ0 = constant, ρ = 0 except when multiplied by g. Thusdefine gρ = b (buoyancy) and then (8) becomes

gρ = b = ρ0N2ζ − (ρ0N

2)zζ2

2+ · · · . (11)

3 Unsteady theory

3.1 Linear theory

Here F1, G1, H1, J1, K1 = 0. Further, we make the Boussinesq approximation, and assumethat N is constant above and below the interface, that is N = N1, z < η and N = N2, z >

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η. Then assume an upward propagating wave packet, a downward reflected wave packetand a transmitted wave packet,

ζ = E {A(T − Z/cg1) exp (−im1z) +RA(T + Z/cg1) exp (im1z)}+ c. c. , z < 0 , (12)

ζ = E S A(T − Z/cg2) exp (−im2z) + c.c. , z > 0 . (13)

where E = exp (ikx− iωt) , Z = εz , T = εt .

and ω =kN1

(k2 +m21)

1/2=

kN2

(k2 +m22)

1/2. (14)

Here 0 < ε � 1 is a small parameter defining the scale of the wave packet. Withoutloss of generality, choose k > 0 so that ω > 0 and the waves propagate in the positivez-direction. Then to ensure that the incident wave and transmitted waves propagateupwards, we must choose m1,2 > 0 as well. X = εx, T = εt (ε << 1) are slow variablesdefining the wave packet and cg1, cg2 are the vertical group velocities, given by

cg1 =ωm1

(k2 +m21), cg2 =

ωm2

(k2 +m22). (15)

We suppose that A(ξ)→ 0, ξ → −∞ so that there are no waves near the interface whenT → −∞. The reflection and transmission coefficients are given by

R =m1 −m2

m1 +m2

, S =2m1

m1 +m2

. (16)

Note that while m1 is required to be real-valued, m2 may be either real-valued or pureimaginary (possible if N2 < N1). In the latter case, let m2 = −iM2,M2 > 0 and thewaves in z > 0 are evanescent. The expression (12) for the incident and reflected wavesstill holds, and |R| = 1. Also formally the expression (13) still holds, but cg2 is complex-valued. Hence A(T − Z/cg2) is a function of a complex-valued variable, and it is thenrequired that A(ξ) be an analytic function of ξ.

3.2 Mean flow

Let 〈·〉 denote an x-average, and set 〈p〉 = p etc. Then the nonlinear mean flow equationsare obtained by averaging equations (1 - 5). In the Boussinesq approximation we get that

ut + 〈uw〉z = 0 , (17)

pz + b+ 〈w2〉z = 0 , (18)

w = 0 , (19)

ζt + 〈wζ〉z = 0 . (20)

bt + 〈wb〉z = 0 . (21)

These are formally fully nonlinear, but it will be sufficient here to work to quadratic orderin wave amplitude. Thus averaging (8) yields

b = g〈ρ0(z − ζ)− ρ0(z)〉 = ρ0N2ζ − (ρ0N

2)z2

〈ζ2〉+ · · · , (22)

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linking b and ζ. Using the linear expressions, (20) can be integrated to give,

ζ = −1

2〈ζ2〉z , so that b = −〈ρ0N

2ζ2

2〉z . (23)

Then (18) can be integrated to yield

p =ρ0N

2

2〈ζ2〉+ 〈w2〉 . (24)

Here an “arbitrary” function of t has been set to zero without loss of generality. Inparticular, (23, 24) show that

p− gρ0ζ = gρ0〈ζζz〉+ρ02N2〈ζ2〉+ 〈w2〉 . (25)

Next, following Sutherland [9], note that the vorticity χ = uz−wx satisfies the equation

ρ0(χt + uχx + wχz) = bx . (26)

Averaging yieldsχt + 〈wχ〉z = 0 , χ = uz , (27)

and then integration with respect to z yields

ut + 〈wχ〉 = 0 . (28)

Here an “arbitrary” function of z is set to zero, since there is no mean flow when thereare no waves, that is as z → ∞ for upward propagating waves. This is equivalent to(17) since it is easily shown that 〈wχ〉 = 〈wuz〉 = 〈wu〉z. But it can now be shown that,correct to second order in wave amplitude on using the linear relation w ≈ ζt,

〈wu〉z = 〈χζ〉t , (29)

and so (17) becomesut + 〈χζ〉t = 0 , or u = −〈χζ〉 . (30)

3.3 Nonlinear theory

Formally, expandζ = αζ(1) + α2ζ(2) + · · · , (31)

where α << 1. Then ζ(1) is given by (12, 13). The second order term will containtwo parts, a second harmonic term proportional to E2 and a mean term, independent ofE, which has been found in section 3. To the required order α2, the averaged interfaceboundary condition (10) is

[p− gρ0ζ]+− =ρ0〈ζ2〉

2[N2]+− . (32)

Noting that w, ζz are continuous across the interface to O(α), we see that the mean bound-ary condition (32) is automatically satisfied by the mean flow solution (25) generated on

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each side of the interface. Note that this results holds for arbitrary N(z) on each side ofthe interface.

In the remainder of this section, we put N = N1,2 on each side of the interface. Itremains to evaluate u from (28), and for this purpose, note that since wave fields describedby (12, 13), χ ≈ −bρ0c ≈ N2ζ/c. Thus, we find that

u =N2

1,2

c〈ζ2〉 . (33)

Hence, when m2 is real-valued, and using (12, 13),

u =2N2

1

c(I2 +R2 + 2IR cos (2m1z)) , z < 0 , (34)

u =2N2

2

cT 2 , z > 0 , (35)

where the incident, reflected and transmitted wave packets are defined by

I = A(T − Z/cg1) , R = RA(T + Z/cg1) , T = S A(T + Z/cg2) . (36)

Note that here A, R, S and hence I,R, T are all real-valued. If instead m2 = −iM2 ispure imaginary, so that the waves are evanescent in z > 0, then the expression (33) stillholds, but (34, 35 ) are replaced by

u =2N2

1

c(I2 + |R|2 + 2I|R| cos (2m1z + φR)) , φR = argR , z < 0 (37)

u =2N2

2 |T |2

cexp (−2M2z) , z > 0 . (38)

We recall that I = A(T − Z/cg1) is real-valued and R = RA(T + Z/cg1), where R iscomplex-valued with |R| = 1 and tan (φR/2) = M2/m1. Also T = SA(T − Z/cg2) whereboth S and cg2 are complex-valued. Note that in both cases the mean flow is discontinuousacross the interface,

[u]+− =2|S|2A2(T )

c(N2

2 −N21 ) . (39)

The origin of this discontinuity is the jump in the vorticity across the interface.There are two main cases of interest. First suppose that the incident wave packet is

localised, for instanceA(ξ) = A0sech(ξ) . (40)

Then the expression (34), or (37), show that there is a localised mean flow associatedwith the incident wave packet, and another with the reflected wave packet, proportionalto A2(T ∓ Z/cg1) respectively, which is a well-known feature of vertically propagatingwave packets. In addition, where these packets overlap briefly near the interface, there isan additional z-dependent mean flow, given by the last term in (34). In z > 0 there is ananalogous localised mean flow in the transmitted wave proportional to A2(T − Z/cg2) inthe propagating case when m2 is real-valued, and a mean flow localised near the interfacein the evanescent case when m2 = −iM2.

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Second, suppose that the incident wave packet is a frontal wave, followed by a uniformwave train, for instance

A(ξ) =A0

2(1 + tanh (ξ)) . (41)

Then for large times, T →∞, A→ 1, and in the propagating case when m2 is real-valued,the mean flow (34) becomes

u→ 2N21A

20

c(1 +R2 +R cos (2m1z)) , z < 0 ,

u =2N2

2A20S

2

c, z > 0 .

(42)

In this case the reflection from the interface has generated a z-dependent mean flow inz < 0, whereas the mean flow in z > 0 is a constant. In the evanescent case whenm2 = −iM2, the mean flow (38) becomes

u→ 4N21A

20

c(1 + cos (2m1z + φR)) , z < 0 ,

u =4N2

2A20

c(1 + cos (φR)) exp (−2M2z) , z > 0 .

(43)

Now there is again a z-dependent mean flow in z < 0, but also a z-dependent mean flowin z > 0 trapped near the interface.

4 Steady case: DJL equation

Here we re-examine the same problem regarded as steady in a frame of reference X =x−Ct, moving with speed C = c+O(α2), where C is the nonlinear wave speed . For thepresent purposes we can use C ≈ c. It is necessary to assume here that after initiationthe wave amplitude A is a constant. It is convenient here to use a stream function ψ suchthat u = ψz, w = −ψx. Then the steady state solution of equations (4, 5) are

ζ − ψ

C= ζ(Ψ) , b = −gρ0(z) + b(Ψ) , Ψ = ψ − Cz . (44)

The vorticity equation (26) reduces to DJL equation

χ = ψzz + ψxx = −b′(Ψ)ψ

ρ0C+ F (Ψ) . (45)

Here the functions ζ(Ψ), b(Ψ), F (Ψ) are unknown. They are usually found from upstreamconditions, for instance, by assuming that ψ → 0, ζ → 0, b→ 0 as x→ −∞. In that case

ζ(Ψ) = 0 , b(Ψ) = gρ0(−Ψ

c) , F (Ψ) = 0 , (46)

and DJL equation (45) reduces to

ψzz + ψxx +N2(z − ψ/C)ψ

C2= 0 . (47)

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However, for periodic waves, this process cannot be carried through. Instead it is notclear how these functions can be determined. It seems that some further information isneeded, such as that from the initial conditions. Thus we can assume that the relation(8) holds in the steady state, that is

b = gρ0(z − ζ)− gρ0(z) , and then b(Ψ) = gρ0(z − ζ) , z − ζ = −Ψ

C− ζ(Ψ) . (48)

Thusb′(Ψ)

ρ0C=N2(z − ζ)

C2(1 + Cζ ′(Ψ)) , (49)

Next for linear steady waves, ζ = ψ/C, and hence ζ(Ψ) is second order in wave amplitude.It follows that DJL equation (45) can be reduced to

ψzz + ψxx +N2(z − ψ/C)ψ

C2= F (Ψ) , (50)

correct to second order in wave amplitude.It remains to determine F (Ψ) which is second order in wave amplitude. Here we use

the mean flow expression (28). First, using DJL equation in the form (50), and correctto second order in wave amplitude

u =N2(z)

C〈ζ2〉 . (51)

Then, second, taking the average of DJL equation (50) yields

uz +N2(z)

C2ψ = F (−Cz) +

(N2)zC〈ζ2〉 , ψz = u . (52)

Since u is given by (51), this determines the functional form of F . Indeed, (52) reducesto

F (−Cz) =N2

C2{C〈ζ2〉z + ψ} . (53)

This analysis shows that the mean flow should not be determined using the steady DJLequation, and instead can only be used for the fluctuating wave components. Instead themean flow is determined from (28).

For example, suppose that N is a constant, and consider a single vertically propagatingwave of constant amplitude A. Then 〈ζ2〉 = 2A2 is a constant, u = 2N2A2/C is a constant,and ψ = uz. It follows that then F (−Cz) = 2N4A2z/C3, and so F (Ψ) = −2N4A2Ψ/C4

is a linear function of Ψ. But when there is also a reflected wave, the situation is morecomplicated, as then u depends explicitly on z, see (42) in the propagating case when m2

is real-valued, and (43) in the evanescent case when m2 = −iM2. Thus, in the propagatingcase, in z < 0,

〈ζ2〉 = 2A20(1 +R2 + 2R cos (2m1z)) , u =

2N21A

20

C(1 +R2 + 2R cos (2m1z)) , (54)

ψ =

∫ z

u dz =2N2

1A20

C{(1 +R2)z +

R

m1

sin (2m1z)} , (55)

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F (−Cz) =2N4

1A20(1 +R2)z

C3+

2N21A

20R(k2 − 3m2

1)

Cm1

sin (2m1z) , (56)

F (Ψ) = −2N41A

20(1 +R2)Ψ

C4− 2N2

1A20R(k2 − 3m2

1)

Cm1

sin (2m1Ψ

C) . (57)

In z > 0, there is just a single vertically propagating wave, and so F (Ψ) is a linear functionof Ψ as describe above, that is in z > 0,

F (Ψ) = −2N42A

20Ψ

C4. (58)

Analogous expressions can be found in the evanescent case, when m2 = −iM2. Thusin z < 0,

〈ζ2〉 = 4A20(1 + cos (2m1z + φR)) , u =

4N21A

20

C(1 + cos (2m1z + φR)) , (59)

ψ =

∫ z

u dz =4N2

1A20

C{z +

1

2m1

sin (2m1z + φR)} , (60)

F (−Cz) =4N4

1A20z

C3+

2N21A

20(k

2 − 3m21)

Cm1

sin (2m1z + φR) , (61)

F (Ψ) = −4N41A

20Ψ

C4− 2N2

1A20(k

2 − 3m21)

Cm1

sin (2m1Ψ

C− φR) . (62)

In z > 0, we have that

〈ζ2〉 = u = 4A20(1+cos (φR)) exp (−2M2z) , u =

4N22A

20

C(1+cos (φR)) exp (−2M2z) , (63)

ψ =

∫ z

u dz = −2N22A

20

M2C(1 + cos (φR)) exp (−2M2z) , (64)

F (−Cz) = −2N42A

20

CM2

(k2 + 5M22 ) exp (−2M2z) , (65)

F (Ψ) = −2N42A

20

CM2

(k2 + 5M22 ) exp (2M2Ψ/C) . (66)

5 Conclusion

In this paper we have re-examined the earlier work by [5] on the reflection of a nonlinearinternal wave incident on an interface separating two regions each of constant but differentbuoyancy frequency. That paper assumed a steady state and so, in the reference framemoving with the horizontal phase speed of the waves, the DJL equation (45) could beemployed. However for the situation, as in [5] and here, when the wave field is periodicin the horizontal direction, certain unknown functions in the DJL equation cannot bedetermined unless some additional information is obtained about the source of the waves.In this paper we show that some knowledge of the wave-induced mean flow allows for thedetermination of a unique DJL equation, see (57) or (62).

Specfically we have reconsidered the steady two-layer problem of [5] by including thetransient development of the waves into a final steady state. Then the wave front generates

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a horizontal mean flow, given by (33) to second order in wave amplitude. This explicitexpression for the mean flow is found by rearranging the usual mean-flow equation (17)into (30) allowing integration in time. The resulting mean flow in each layer is givenby (34) and (35). To achieve this same mean flow from the steady DJL equation, anadditional assumption (48) must be made, which is the requirement that there is noaverage mass flux through a horizontal surface.

The previous work [5] used a different DJL equation, in which the arbitrary func-tions were obtained with reference to a basic state without any waves. In effect thatcorresponds to a different source of the waves than that used here. In [5] the dynamicinterfacial conditions were found to generate a contribution to the mean flow due to aninhomogeneity at second-order in wave amplitude in the dynamic interfacial boundarycondition. The present work demonstrates that the mean pressure field by itself balancesthis inhomogeneity.

The mean flow in the final steady wave solution is shown here to be discontinuousacross the interface. This discontinuity is present whether the waves in the upper layerare transmitted, as at the tropopause or the mesopause, or evanescent, as they may be atthe stratopause. The strength of the jump in the mean flow is proportional to the jumpin the square of the buoyancy frequency. This important result provides a likely scenariofor elevated levels of turbulence that appear to be present at the tropopause, and suggeststhat similar patterns of turbulence are present at the stratopause and mesopause.

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