18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
Internal wave breaking depending on stratification
Sergey Kshevetskii, Sergey Leble
I.Kant Russian State University
Theoretical Physics Department
Al.Nevsky street, 14, Kaliningrad, Russia
renger@mail.ru
Abstract :
Euler equations for incompressible fluid stratified by a gravity field are investigated. It is found out that the system
of Euler equations is not enough for statement of a correct generalized problem. Some auxiliary conditions are
offered and justified. A numerical method is developed and applied for study of processes of whirl destruction
and mixing in a stratified fluid. The dependence of vortex destruction on a stratification scale is investigated
numerically and it is shown that the effect increases with the stratification scale. However, the effect of vortex
destruction is absent when the fluid density is constant.
Key-words :
stratification; turbulence; mixing
1 Introduction
Internal wave breaking is one of interesting nonlinear effects, which is characterized by disin-
tegration of waves and by formation of typical spots with an intensive small-scale convection
inside them. The spots are often termed convective or turbulent ones because the convection in-
side them looks like turbulence. Static stability in the fluid becomes recovered with the course
of time, but the density inside the spot remains different from the surrounding stratification.
The phenomenon of internal wave breaking going with formation of a convective spot often
is termed "internal wave mixing". The effect of internal wave breaking is often observed in
the ocean Miropolskii (1981). The phenomenon is registered in the atmosphere Pfister et al.
(1986). The broken down internal waves reorganize stratification with time. Therefore, it is im-
possible to understand and explain stratification of the ocean or the atmosphere without taking
into account the effect.
In McEwan?s experimental papers, the phenomenon has been studied by means of thin laser
measurements. Internal waves were excited in a laboratory tank with sizes of 25cm ?50cm ?
25cm filled with a stratified fluid. Stratification is formed by dependence of water saltiness on
height. The ratio ? = ??? = 0:04, so the stratification is a weak one. Observation of small-scale
structures had been based on dependence of refraction index of light on liquid density. McEwan
has discovered and studied different stages of evolution of wave breaking and mixing McEwan
(1971), McEwan (1983): overturning, development of interleaving microstructure, restoring of
static stability.
Very large density gradients and velocity shears eventually are formed by mechanisms of the
effect under study. These large density gradients essentially influence dynamics. The Boussi-
nesq approximation is admissible before vortex destruction, but in time it influences the small-
scale convection generated, and is not used. Also, it is wisely to presume the solution may
contain discontinuities. It seems to be of interest to study nondifferentiable, generalized solu-
tions of hydrodynamic equations of a stratified incompressible fluid placed into a gravity field.
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
2 Basic system of equations
We suppose the fluid behavior is described by Euler equations for an incompressible fluid
@?
@t +
@?u
@x +
@?w
@z = 0;
@?u
@t +
@?u2
@x +
@?uw
@z = ?
@p
@x; (1)
@?w
@t +
@?uw
@x +
@?w2
@z = ?
@p
@z ??g;
@u
@x +
@w
@z = 0.
Here ? is the density, p is the pressure, u and w are the horizontal and vertical mass velocities
of the fluid, t is time, x;z are the horizontal and vertical coordinates respectively, g is the free
fall acceleration.
It is supposed as well that the unperturbed fluid is stratified in density exponentially: ?0(z) =
?00 exp?? zH?: Here H = 6:23m;?00 = 1000kg=m3. The chosen value of H approximately
corresponds to the one for McEwan?s experiments McEwan (1971), McEwan (1983).
The boundary condition is (v;n) = 0 at the boundary @? of the domain ?, where v = (u;w)
and n is a normal line to the boundary. Keeping in mind McEwan?s experiments, we take a
rectangle with horizontal dimension 50 cm and vertical dimension h = 25 cm as the domain ?.
3 Statement of the generalized problem
Theorem 1 Let a solution to equations (1) be differentiable. Then the functional
Hnonl =
Z
?
?
? u
2 +w2
2 +?gz + +gH
? ln
?
?0 (0)
?
+ (?0(z)??)
??
d? (2)
is conserved over solutions. The integrand in (2) is strictly non-negative at ? ? 0. Functional
(2) turns into the functional of wave energy from a linear theory Miropolskii (1981), when we
take a small-amplitude limit.
Let us generalize (2) to a case when solution may be nondifferentiable. The functionals
of mass, hydrodynamic energy should be conserved over nondifferentiable solutions; it fol-
lows from their physical sense. However the functional R
?
h
gH
?
? ln
?
?
?0(0)
??i
d? is not
conserved in case of discontinuities in the density. We demand
dHnonl
dt ? 0: (3)
The sign "?" is chosen because conservation of (2) is impossible in case of discontinuities in
density, but ">" in (3) leads to an unstable problem. We will use (2) and (3) as a basis of the
theory of generalized solutions to nonlinear equations.
From the physical point of view, the set of equations (1) is incomplete because does not
include the energy conservation law. Let ! ? ? be an arbitrary star domain with piecewise
smooth boundary, and let S(!) be a boundary surface of the domain !, dS = dS n, where dS
is a surface element of S(!), and n is a vector of an outer normal line to S(!). We write an
integral relation for energy
Z
!
e(x;z;t)d??
Z
!
e(x;z;0)d???
tZ
0
I
S(!)
e(x;z;t)v(x;z;t)dSdt = 0; (4)
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
Here e = ?
?
u2+w2
2 +gz
?
. If the solution is differentiable, then (4) follows from (1); but if
the solution does not belong to a class of differentiable functions, then relation (4) does not
follow from (1). Therefore, permitting nondifferentiable solutions, we have to include (4) into
the system of simulative equations as an individual equation.
Relations (1), (4) are not enough for statement of a correct problem. (3) should be fulfilled
for stability of the nonlinear problem. As conservation of energy functional (4) is postulated,
the relation @
@t (f (?))+r(f (?)v) = ?; (5)
follows from (3) and (4). Here f (?) = ?ln(?), ? ? 0. Requirement of compatibility of
(5) with the equation of continuity in (1) results in conclusion that ? may be nonzero only at
discontinuities. Evidently, ? is unknown. Coming from physical reasons, one can suggest to
find ? from the condition of minimum of j?j under the condition ? ? 0.
Using the set of equations (1), we routinely build some integral relations for definition of a
generalized solutionZ
?
?s(x;z;0)?(x;z;0)d??
Z
QT
?@?s@t + +?u@?s@x +?w@?s@z
?
dQT = 0; (6)
Z
?
u(x;z;0)?(x;z;0)us(x;z;0)d?+
Z
?
w(x;z;0)?(x;z;0)ws(x;z;0) d?+
Z
QT
?
?u @us(x;z;t)@t +?u2 @us(x;z;t)@x + +?uw @us(x;z;t)@z
?
dQT
+
Z
QT
?
?w@ws(x;z;t)@t +?uw@ws(x;z;t)@x + +?w2@ws(x;z;t)@z +?gws(x;z;t)
?
dQT = 0:
Relations (6) do not contain pressure p(x;z;t). To calculate p(x;z;t), it is possible to use
an individual equation giving p(x;z;t) through v(x;z;t).
However, the above integral relations are not enough for unambiguous definition of a gen-
eralized solution. To be sure, we should add relation (4) and equation (5) in an integral form to
equations (6).
We construct here the definition of a generalized solution, which should be acceptable with
physical point of view and correct with mathematical point of view. From the previous re-
viewing, it is evidently that (6) is not enough to find unique a physically justified generalized
solution. If the solution is piecewise-continuous and restricted in QT, then all integrals in (4)
exist; and from (5), (4) it followsZ
?
e(x;z;t) d? =
Z
?
e(x;z;0) d?; (7)
Z
?
f(x;z;t1)d? ?
Z
?
f(x;z;t2)d? t1 > t2: (8)
Definition 2 Let v(x;z;t) = (u(x;z;t);w(x;z;t)). We term ?(x;z;t) =
?(x;z;t)
v(x;z;t)
?
a
generalized solution of equations (1), if u(x;z;t) = @?(x;z;t)@z , w(x;z;t) = ?@?(x;z;t)@x , ?j@? = 0
and if (7), (8), (6) are fulfilled for any ?s(x;z;t) 2 C10 (?) ? C1[0;T], us(x;z;t) = @?s@z ,
ws(x;z;t) = ?@?s@x , ?s(x;z;t) 2 C10 (?)?C1[0;T], ?s(x;z;T) = ?s(x;z;T) = 0.
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
4 Simulation of vortex destruction
We solve Euler equations (1) by finite-difference method. The numerical method developed
allow simulation of nonsmooth, generalized solutions, because finite-difference equations used
satisfy all conditions of (7), (8), (6), 3 at h ! 0 Kshevetskii (2006).
We take the initial condition:
?(x;z;0) = Ae
n
?
h(x?x
0
lx )
2+(z?z0
lz )
2io
; (9)
?(x;z;0) = ?0 (z), where lx = 0:16 m, lz = 0:052 m, A = ?0:0095 m2=s, x0 = 0:5 m,
z0 = 0:125 m. This initial condition approximately corresponds to the disturbance created
from an oscillating paddle in the tank in McEwan?s experiments McEwan (1983), McEwan
(1971). The horizontal scale is approximately twice more than the vertical one. ?mplitude of
the vertical velocity is of 5 cm/s, amplitude U of the level velocity is of 16 cm/s. A scale
of the initial wave l = h=(2?) = 0:04m. The Froude number characterizes the degree of
nonlinearity, Fr = U=(Nl) = 3, the Richardson number is Ri =
?
?1? d?d
?
=?@U@z?2 = 0:1,
where N = pgH = 1:2s?1. Known necessary condition of flow instability is Ri < 0:25.
The fluid density for t = 3s, 7s is shown in fig. 1, 2.
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0.05
0.1
0.15
0.2
960964
968972
976980
984988
992996
1000
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0.05
0.1
0.15
0.2
960964
968972
976980
984988
992996
1000
Figure 1: Fluid density at t = 3s. H = 6:23m. Figure 2: Field density at t = 7s. H = 6:23m.
The overturning and formation of small-scale structures is forestalled by formation of a
tongue of a heavy liquid, penetrating into strata of a light liquid, and on the contrary. Some
layer structure containing segments with inverse density is generated as a result. At t = 5s,
abruptions arise in tongues, and isolated fragments of a light liquid arise inside a heavy liquid,
and on the contrary. At t = 7s, the process of breaking becomes intensive. At t = 9s, about
10% of fluid in the tank is retracted in intensive small-scale convection. The vortex is atomized
and the convective spot slowly grows. However, with the course of time, small-scale blobs of
a heavy liquid subside downwards, and blobs of a light fluid go upward. As a result, stable
stratification is practically restored to the moment t = 14s, but not for smooth density, and for
density being averaged with the scale larger the scale of small-scale heterogeneities.
As a whole, the picture of wave breaking qualitatively coincides with circumscribed in McE-
wan (1971), McEwan (1983). However, the last stage of reestablishing of continuous stratifi-
cation is absent. It is because the dissipative effects are not taken into account in the model.
At the first stage, the disturbance behavior is typical for internal waves and is perfectly
explained by the theory of small-amplitude internal waves: left-hand and right-hand waves arise
from this vortex. Because of the initial density perturbation is equal zero, the field of density
perturbation is antisymmetric, and the field of a flow function is centrally symmetric. The
symmetry is maintained in good approximation even when irregular movement is developed.
Irregular structures in the flow function appear later, than in the density. Probably it is
explained by the fact that a flow function is as an integral of velocity, and is the most smooth
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
of considered physical fields. Two opposite jet flows are formed at t = 3 s, and then they in
unison create considerable velocity shift. The energy of small-scale waves is scooped from a
kinetic energy of the vortex due to such a mechanism of instability development. It explains
great intensity of the developed small-scale convection.
4.1 Investigation of dependence of vortex destruction on a stratification scale
Identical simulations for stratifications with doubled and diminished twice H have been carried
out for investigation of dependence of phenomena on a stratification scale. Outcomes for H =
3:1m are shown in fig. 3. It is the case of Fr = 2, Ri = 0:22. We see that diminution of H
essentially affects wave process. Fluid movement becomes more horizontal. The effect of wave
breaking is starting later in spite of the fact that the inverse Vaisala-Brunt frequency is less. The
wave breaking develops more slowly, and destruction runs languidly. Regular wave motion of
the fluid is maintained as a whole, and only separate "winged nuts" are generated.
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0.05
0.1
0.15
0.2
920925
930935
940945
950955
960965
970975
980985
990995
1000
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0.05
0.1
0.15
0.2
976978
980982
984986
988990
992994
996998
1000
Figure 3: Fluid density at t = 7s. H = 3:1m. Figure 4: Fluid density at t = 7s. H = 12:4m.
In fig. 4 the outcomes of simulation of the same wave, but for the stratification with doubled
value of H, is shown. It is the case of Fr = 4, Ri = 0:1. Comparison with simulations
relating to normal value of H and with simulations relating to diminished twice H displays
that the phenomenon of vortex destruction, intensity of generation of small-scale convection is
increased as H.
Because we have discovered the destruction effect increases with H, we have carried out
simulation of evolution of a starting vortex for stratification with very large H = 311;5m. The
considered medium is almost homogeneous: ? = ??? = 0:0008. Now we consider the case
of Fr = 15, Ri = 0:004. The simulation outcomes are shown in Fig. 5. We see the wave
collapses, and the time of starting of wave breakdown is the same as in case H = 12:4 m.
Therefore, for large H, the time of wave breakdown starting is determined not so much by
value of H, how much by starting conditions. By considering the flow function, we discover
formation of oppositely directed jet flows. Originating of this velocity shears explains, probably,
the instability development. Small-scale fluctuations appear in the density field, and with some
retardation, they appear in the flow function.
Some late instant of evolution of the flow function for the same initial vortex, but in the fluid
of a constant density, is shown in fig. 6. We see that the whirl breakdown and formation of
small-scale convection is absent.
Numerical simulations reveal that the phenomenon of breakdown becomes more and more
brightly with increase of stratification scale H. Nevertheless, the phenomenon of vortex de-
struction is absent, when the fluid density is strictly a constant. It points out that continuous
limit from a stratified fluid into the case of a fluid of constant density is absent for our initial
conditions. Approximation of density by a constant is out of physical sense in such cases, even
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
0.05
0.1
0.15
0.2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.05
0.1
0.15
0.2
0.25
Figure 5: Lines of the flow function at t = 7 s.
H = 311:5.
Figure 6: Lines of the flow function at t = 7 s in
a homogeneous liquid.
if the density varies very little.
5 Conclusions
The Euler equations for an incompressible stratified fluid were studied. Non-negative nonin-
creasing functional extending wave energy functional of the theory of linearized Euler equations
is suggested. The functional value is conserved over differentiable solutions, and diminishes, if
density discontinuities are present. Functional properties have allowed using them for analysis
of correctness of the generalized problem. It is shown that Euler equations are not enough for
statement of a correct generalized problem for a stratified fluid. One auxiliary condition is an
energy conservation law, and the second is a special condition for density (5). The statement
of a generalized problem is formulated. The phenomenon of destruction of starting vortex in
conditions of McEwan?s laboratory experiments McEwan (1971), McEwan (1983) has been
simulated and studied. Qualitative concurrence of the simulated effect with the one observed in
laboratory experiments is good, but the last stage of restoring of smooth stratification is absent
due to usage of an ideal fluid model. By means of numerical experiments, dependence of the
solution on stratification scale H is studied. It is revealed that the effects of vortex destruction
and formation of small-scale convection increases with H for strongly nonlinear waves. On
the contrary, the effect weakens with diminution of H. The effect is not observed, if the fluid
density is strictly a constant.
References
Miropolskii, Ju.Z. 1981 Dynamics of internal gravity waves in the ocean. Leningrad, Gidrom-
eteoizdat.
Pfister,L. Starr, W., Craig, R., Lewenstein, M., Legg, M. 1986 Small-scale motions observed by
aircraft in the tropical lower stratosphere: evidence for mixing and its relationship to large-
scale flows. Journal of Atmospheric Sciences. 43 pp. 3210-3225.
McEwan, A.D. 1971 Degeneration of resonantly-excited standing internal gravity waves. J.
Fluid Mech. 50 pp. 431-448.
McEwan, A.D. The kinematics of stratified mixing through internal wavebreaking. 1983
J.Fluid Mech. 128 pp. 47-57.
Kshevetskii, S. 2006 Study of vortex breakdown in a stratified fluid Computational Mathemat-
ics and Mathematical Physics. 11 pp. 1988-2005.
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