Kelvin-Helmholtz Instabilities
Lew Gramer
Lew.Gramer@noaa.gov
GFD-II
Friday, April 27, 2007
GFD-II 2007: Kelvin-Helmholtz Instabilities
Table of Contents
1. Introduction: The nature of the problem........................................................................................3
Focus of this paper...................................................................................................................... 3
Further reading............................................................................................................................ 4
2. Scale of Kelvin-Helmholtz instability ...........................................................................................4
Vertical scales must be large enough…...................................................................................... 4
And lateral scales must be small enough… ................................................................................ 5
But lateral scales must not be too small….................................................................................. 5
3. Sufficiency of two dimensions: Squire’s theorem .........................................................................6
4. Simple conditions for, and limits on shear instability....................................................................6
The role of buoyancy: Helmholtz two-layer system................................................................... 6
An extreme case: The “vortex sheet” ......................................................................................... 8
The Bernoulli equation: condition for parallel flow stability ..................................................... 9
Evolution into the continuous case ........................................................................................... 13
5. Continuous profiles: the Taylor-Goldstein equation....................................................................14
Necessary condition: Richardson number criterion.................................................................. 16
Why ¼? (Secondary instabilities, turbulence, and mixing) .................................................. 17
Finding sufficient conditions: empirical studies ................................................................... 18
Limits to growth: Howard’s semi-circle theorem..................................................................... 21
6. Meditations on the Taylor-Goldstein equation ............................................................................22
The role of viscosity: the Reynolds number ............................................................................. 22
Comparison of barotropic and Kelvin-Helmholtz instability ................................................... 23
Dynamic similarity: lateral and vertical instability................................................................... 23
7. Importance of Kelvin-Helmholtz instability................................................................................25
Surface gravity waves............................................................................................................... 25
Role in transferring momentum from wind to current.......................................................... 25
Homogenization – ABL and oceanic ML................................................................................. 26
Monin-Obukhov depth scale................................................................................................. 26
The “k parameterization” problem ........................................................................................... 27
Conclusion .......................................................................................................................................28
Acknowledgements..........................................................................................................................31
References........................................................................................................................................31
Lew Gramer
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GFD-II 2007: Kelvin-Helmholtz Instabilities
1. Introduction: The nature of the problem
The topic of this paper is the Kelvin-Helmholtz instability, an instability that arises in parallel
shear flows, where small-scale perturbations draw kinetic energy from the mean flow. This is
inherently a small-scale, irrotational phenomenon, as we will discuss at length below. All of the
dynamics that we have studied to date in GFD have been at scales where the Rossby number is
small to very small, and thus where rotation is inherently important. However, the dynamics of
such larger scale motions in the real world may still depend in critical ways upon smaller scales,
where rotation is less important.
For this reason, it is important in the context of geophysical systems at least to understand the
basic dynamics of small-scale flow as well. This is particularly likely to be true of small-scale
motions where there is instability – for intuition tells us that these are precisely the motions that
are most likely to modify meso- and planetary-scale dynamics in important ways. And in fact,
intuition may even lead us to suspect that small-scale instabilities may play a role in the forcing of
larger systems in the real world. As we shall see, both of these intuitions are in fact correct.
It is also worth mentioning that the consideration of small scales is an attractive topic in itself. For
one thing, it obviates the need to cast our equations of motion in a rotating coordinate frame,
greatly simplifying their manipulation. And it leads to another advantage also. We are generally
taught by GFD to mistrust conclusions of laboratory experiment and normal physical intuition, as
being fundamentally unable to reproduce some of the most important phenomena of large-scale
motion. Yet small-scale dynamics not only allows us to use direct experiment. It actually requires
us to do so, if we are to derive some of the most important results. And as a corollary, small scales
permit us to find very intuitive examples, as we will see.
Focus of this paper
The term Kelvin-Helmholtz was originally applied to a particular set of gravity-wave phenomena
at discontinuities, originally investigated by von Helmholtz in 1868. Over time, this term has come
to refer to a broader class of unstable small-scale motion – including some phenomena that we
know do occur in the real ocean and atmosphere. We follow this more general usage in this paper.
We must also at the outset distinguish Kelvin-Helmholtz instability from true (small-scale)
turbulence. These two phenomena are not truly separable, and in fact, the two play a complex and
interactive role in the earth system’s dynamics. However, KHI is a phenomenon that can be
adequately considered in two dimensions, as we will see, while turbulence is an inherently threedimensional phenomenon, and therefore demands a more extensive, sophisticated analysis.
Further, our treatment will always assume a basic state of static stability, i.e., that heavier fluid in
general lies below lighter fluid. This assumption may not precisely hold in all real geophysical
situations, for example in regions of very rapid atmospheric heating from below, or where a fast
ocean current interacts with a less dense water mass along a coastal front. However, it does hold in
many situations of oceanographic and meteorological interest. And it allows us to ignore the
complexity of a phenomenon known as Rayleigh-Taylor instability, when the complexity of the
Kelvin-Helmholtz variety is already quite sufficient! So we will choose to assume static stability
in all circumstances here. Along the same lines we will sometimes assume – without stating
explicitly – that all fluids across a domain are chemically and mechanically miscible. This may be
Lew Gramer
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GFD-II 2007: Kelvin-Helmholtz Instabilities
a poor assumption, particularly in the case of ocean wave production by wind. However whenever
this assumption may limit important results, I’ll try to indication this assumption explicitly.
Finally, the class of motions we choose to study here must be distinguished from the instabilities
resulting from an obstruction (e.g., a rough boundary) in a fluid flow. Karman vortex streets and
other such phenomena are certainly significant – not only in the laboratory, but also at lateral and
vertical boundaries in real geophysical flows. However, they are dynamically distinct from the
Kelvin-Helmholtz instability, which by contrast can theoretically occur across any sheared region
within a fluid, and so these phenomena will not be considered here.
Further reading
The original literature on general Kelvin-Helmholtz instabilities stretches back some 150 years.
The most important results are attributed to seminal papers of the last half-century – e.g., Miles
(1961), Howard (1961), Klebanoff et al (1962), and Thorpe (1971). Some of these authors were
applied mathematicians, using their own particular notation, and many of their derivations can be
extremely difficult to follow now. However the subset of these results that we will try to present
here is derived in detail, in sections 11.6 through 11.13, of Pijush K. Kundu’s classic Fluid
Mechanics (1990). A less rigorous, more intuitive treatment, but one that helps place these results
in the context of general geophysics can be found in Cushman-Roisin (1994), sections 11.1 to 11.3.
And further applications for KHI in modern geophysical modeling research is found in the
upcoming 2nd edition of that eminent text, Cushman-Roisin and Beckers (2007), chapter 14.
2. Scale of Kelvin-Helmholtz instability
Vertical scales must be large enough…
All the derivations we perform below ultimate rely on regions of sufficiently large vertical extent.
We acknowledge however, the possibility that boundaries may play a critical role in the dynamics
of small-scale instabilities. An upper boundary (e.g., the free surface of an ocean, for internal KHI),
as well as a lower boundary (a flat bottom – or sloped, as will be the case near the wave breaking
zone on a coast) is likely to have sometimes a stabilizing, and perhaps sometimes a destabilizing
effect on perturbations at a velocity shear. Thus a full treatment of small-scale instability would
necessarily have to consider “shallow waves”, as well as complex three-dimensional instabilities.
In effect, we would abandon the simple focus we argued for in the introduction! However we are
fortunate. For as we will see in the final section of this document, some of the most important realworld applications of KHI occur in regions where we may assume there is no upper or lower
boundary. One topic where consideration of small-scale instability is critical is the development of
surface waves at the air-sea interface in the open ocean – where we may certainly consider the
extent of both sub-domains (air and sea) to be effectively infinite.
And as we’ll briefly mention below, another topic where KHI plays a key role is in finding
appropriate parameterizations for viscosity in eddy-resolving models of the ocean and atmosphere
circulation. Yet here also, away from the narrowest coastal regions of western boundaries, we may
hope to be able to allow vertical scales large enough to examine the development of KelvinHelmholtz instability, without considering boundaries. As such, we will choose to leave a full
treatment of unstable flows within vertically bounded regions – for example, examining the role of
inflection points within a mean flow along a theoretical pipe – to the textbooks on fluid mechanics.
Lew Gramer
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GFD-II 2007: Kelvin-Helmholtz Instabilities
And lateral scales must be small enough…
Horizontally coherent, nearly vertical motions like that pictured on our title page and elsewhere,
are observed to occur at very small scales in the real atmosphere and ocean. What is more,
experiments over many decades have shown that wave-like instabilities in mean flows can easily
be produced that are dynamically identical to these real-world phenomena, but under laboratory
setups that have been carefully scaled and controlled to eliminate rotation. We surmise from these
two facts, that such instabilities can be driven by very small-scale dynamics, without any reference
at all to planetary sphericity or diurnal rotation.
We will exploit this observation, to study how such instabilities develop from a purely irrotational
system. This is why we made clear in the introduction, that an essential assumption in the
development that follows, will be that our motion will consistently remain irrotational – both
before and after modification by a perturbation. Therefore, in a real atmosphere or ocean on a
rotating planet, this clearly restricts us to characteristic horizontal scales that are much smaller
than the effective Rossby radius appropriate to the region of consideration: L << Rd*.
But lateral scales must not be too small…
We have said we will only consider very small scales. And yet throughout this paper, we will
choose to ignore the direct effects of such small-scale phenomena as molecular viscosity, surface
tension or density diffusion. How can this be reconciled? First, diffusion by molecular processes
common in either air or ocean is a slow process over macroscopic scales. Thus for the time scales
of real Kelvin-Helmholtz instabilities, we will assume a priori that we may ignore its effect.
Second, with respect to surface tension: we will see in a later section that the equation we find to
relate a perturbation’s surface displacement to its time evolution is a simple first-order one. Yet
surface tension introduces an additional term into this equation, which is of second order in
displacement. The net effect of this change is to modify the effect of gravity in the dispersion
relation between frequency and wavelength: In effect, surface tension acts as an additional
restoring force in wave dynamics. Interfacial waves where molecular surface tension plays an
equal or dominant role relative to gravity are generally referred to as capillary waves.
These waves are the inevitable first “fillip”, likely to precede the development of some largerscale instabilities. However, it can be shown (Kundu sec. 7.7) that for perturbations of wavelength
greater than a certain small value (λ ≈7cm for example, at the air-sea interface), Kelvin-Helmholtz
instabilities can still progress, while surface tension may be ignored. Thus for fluid interfaces, we
must limit our study below to the further development of instabilities after an initial unstable
perturbation has developed. By this assumption we may ignore capillary waves from this point
forward. Yet it will be worthwhile for the reader to bear in mind this lower limit on wavelength
(i.e., upper limit on wavenumber), when we derive criteria for wave instability in section 4 below.
Finally, molecular viscosity is likely to play an important role in small-scale instabilities: below
we will try to decompose its stabilizing or destabilizing effect in the absence of stratification.
Lew Gramer
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GFD-II 2007: Kelvin-Helmholtz Instabilities
3. Sufficiency of two dimensions: Squire’s theorem
In the examples throughout this paper, we ignore the second horizontal dimension. We are in fact
about to derive a set of bounds and criteria for instability in small-scale perturbations, relying on
two-dimensional governing equations. How can we be sure that any bounds or conditions on
instability that we derive from such an analysis will also hold in real, three-dimensional flows?
For this, we rely on an important and striking result known as Squire’s theorem (Squire 1933).
This theorem actually relies on a coordinate transform in wavenumber space. The result is a
simplification of the normal mode analysis, which Squire uses to show that – for each unstable
three-dimensional wave solution to a perturbed system, there must always exist a two-dimensional
wave solution which is unstable at higher wavenumber. In other words, a two-dimensional system
is always more unstable than any equivalent three-dimensional system, using the same analysis!
The transformation is a brilliantly simple one, with pure horizontal wavenumber for the 2-D
system defined by the transformation κ = (k2 + m2)1/2, while total (complex) phase speed c remains
the same. Squire demonstrates that in the Flatlandian system, unstable perturbations of the
transformed two-dimensional momentum equations will grow as exp(κci), whereas threedimensional disturbances are confined to grow at the rate kci, where by definition kci ≤ κci.
Squire’s theorem tells us that KHI and, as we will see, pure barotropic instability also, can be fully
described and analyzed using equations in two dimensions. It is important to note in passing what
this means: for not only is this a trick for simplifying stability analysis. Rather, the theorem
defines for us a dynamic similarity, such that we may say in effect, that all instabilities in parallel
mean flows are inherently two-dimensional. We will analyze the implications of this below.
4. Simple conditions for, and limits on shear instability
The role of buoyancy: Helmholtz two-layer system
We will follow Cushman-Roisin, by first doing a simple analysis of the classic two-layer system
based on energetics, to derive a natural condition that must me met if instability in this system can
possibly lead to mixing. We’ll see that here, as above, instability is always possible for sufficiently
short perturbation scales. (This statement however, may be less complete than we wish. To wit,
see our earlier discussion on the restoring – i.e., instability-dampening – role of surface tension.)
In this section, we consider a two-dimensional domain infinite in vertical and horizontal extent.
This two-layer system is essentially that which was considered by Hermann von Helmholtz.
We first posit a priori that conditions do exist which allow instabilities to derive energy from the
shear (U2-U1) in our mean flow. We further consider that these instabilities ultimately lead to
mixing of the fluid over some finite vertical distance ∆H in the domain, centered on the initial
discontinuity. This in turn results in a net gain in potential energy within the mixed region. The
resulting mixed region is shown in the figure below, adapted from Cushman-Roisin 2007.
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GFD-II 2007: Kelvin-Helmholtz Instabilities
Time Æ
We then ask what conditions must be met by the resulting energy balance, in order for us to
validate these assumptions. (Note this differs trivially from the treatment in Cushman-Roisin, only
in that we do not assume vertical boundaries to our system. These in fact prove to play no role in
the dynamics of the developing instability, under our energetic analysis.)
We may characterize the available potential energy in our initial or basic system by the integrals:
∆H / 2
∆H
0
∆H / 2
∫ ρ 2 gz ⋅ dz +
∫ ρ1 gz ⋅ dz = 12 ρ 2 g
∆H 2
∆H 2 1
∆H 2 1
3∆H 2
− 0 + 12 ρ1 g
− 2 ρ1 g ⋅ ∆H 2 = 12 ρ 2 g
− 2 ρ1 g
4
4
4
4
From this, based on our assumption of mixing over the length ∆H, we develop into a final state
with the following net gain in potential energy ∆PE = 18 ( ρ 2 − ρ1 ) g ⋅ ∆H 2 . We understand that the
net gain in PE can only be accomplished in this (or any similar) system by a corresponding loss in
kinetic energy from the sheared mean flow. By a similar vertical integration across ∆H for our
initial and final states, this minimal conversion of kinetic energy to potential energy is found to be
∆KE = 18 ρ (U 2 − U 1 ) 2 ∆H . Here we define ρ = ( ρ 2 + ρ1 ) / 2 , U = (U 2 + U 1 ) / 2 , and we have
further assumed per Boussinesq that ρ ≈ ρ1 ≈ ρ 2 for simplicity.
For mixing to occur – as we propose that it inevitably must from an unimpeded unstable
perturbation – our simple energetic analysis leads thus to the following necessary condition:
( ρ 2 − ρ1 )
ρ
g ⋅ ∆H ≤ (U 2 − U 1 ) 2
(4.0)
It is important to be clear that this inequality represents an upper bound on the transfer of mean
flow kinetic energy required to achieve fluid mixing. As we will see later in the section on the
Richardson number criterion, the inequality above clearly does not represent a least upper bound
on that energy requirement. The weakness of this result can partly be attributed to the unrealistic
nature of the inviscid, narrow scale-range discontinuities we are considering here; this point is
nicely illustrated by the pathological example in the next sub-section.
However, this weakness is really due to the fact that mixing, under the assumptions we’ve made, is
not adequately explained by two-dimensional phenomena like KHI – even if they are allowed to
become fully non-linear. We will hopefully discuss this further, albeit briefly in the final section.
Lew Gramer
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GFD-II 2007: Kelvin-Helmholtz Instabilities
An extreme case: The “vortex sheet”
A curious conclusion can be drawn from inequality (4.0). Imagine a simple system where a plane
discontinuity separates two regions of parallel flow, both of the same density. In this scenario, if
mixing occurs then the loss of potential energy will actually be zero. This implies that the
inequality above will always be satisfied. Any perturbation in the planar discontinuity however
slight should draw sufficient kinetic energy from the shear discontinuity to develop indefinitely.
This is especially so for non-horizontal plane discontinuities – where we may expect any potential
energy barrier to shear instability to be even smaller with increased angle from the vertical.
Note that this places no constraints on the transfer of kinetic energy from the sheared mean flow
either. In fact, we might presume that in the absence of momentum diffusion, the instability and
resulting mixing would in fact cause no net change in total kinetic energy of the system at all!
Naturally though, as the instability develops, transitions to secondary instabilities, and ultimately
to turbulence and mixing, the scales of the motion (and of the energy of the system) would change.
To continue our discussion, we have just concluded that where a fluid has no variation in density
within a region, then instability will theoretically always be present where ever there is a shear
discontinuity. This simple setup is known as a “vortex sheet” for obvious reasons, and represents
an extreme case in the analysis of Kelvin-Helmholtz instability. Yet consider that homogenous
water columns, over small scales at least, are actually quite common in the ocean: indeed they are
a natural result of the very dynamics we study in this paper!
Why then do we not observe “vortex sheets” wherever there is shear in the mean flow over a
homogeneous region? To answer this question, we need only recall our decision at the outset, to
ignore surface tension and boundaries, and to decouple viscosity from the effects of buoyancy. All
of these factors can set constraints on the development of small-scale instability, even where
buoyancy is minimal. And as this argument implies, I believe these effects must play quite a
significant role in the real world, particularly in the air-sea interface and oceanic mixed layer.
Lew Gramer
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GFD-II 2007: Kelvin-Helmholtz Instabilities
The Bernoulli equation: condition for parallel flow stability
We continue with the two-layer system, and seek to derive appropriate condition for instability by
considering a wave-like perturbation at a discontinuous interface between two fluids, as a solution
to Euler’s irrotational momentum equations. This setup is summarized by the figure below:
Figure 1: Kelvin-Helmholtz instability developing from a wave-like perturbation of wavenumber k at the
interface between two vertical regions (a); resulting in breaking (b), and ultimately in mixing like that which
we assumed a priori in the previous section. (Figure from Cushman-Roisin and Becker 2007.)
We develop this example at length, as it is intuitive and easy to manage. Further it will prove to be
an illustrative preliminary when we consider the more realistic case of a continuous distribution of
both velocity and density in the sections below on the Taylor-Goldstein equation and the
Richardson number criterion. For we will find that that famous criterion begins from essentially
the same equations, subject only to somewhat different boundary conditions across the interface.
We have assumed that our horizontal domain scale is small relative to the Rossby radius; a motion
uniform over a domain, whose shear is zero, may be considered to have zero absolute vorticity as
well. This condition allows us to characterize the momentum balance in our two-dimensional
system using the relatively simple Euler equations:
∂u
∂u
∂u
1 ∂p
+u
+w
=−
,
∂t
∂x
∂z
ρ ∂x
Lew Gramer
∂w
∂w
∂w
1 ∂p
=−
−g.
+u
+w
∂z
∂t
∂x
ρ ∂z
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(4.1)
GFD-II 2007: Kelvin-Helmholtz Instabilities
For barotropic motions (ρ ≡ ρ(p)) like that of our basic state, these momentum equations lead to
the following system of equations for the lateral and vertical energy balance of our system:
r
∂u ∂ ⎡ 1 2
dp ⎤
+ ⎢ 2 q + ∫ ⎥ = (u × ζ ) x ,
∂t ∂x ⎣
ρ⎦
⎤
r
∂w ∂ ⎡ 1 2
dp
+ ⎢2 q + ∫
+ gz ⎥ = (u × ζ ) z . (4.2)
∂t ∂z ⎣
ρ
⎦
For scales much less than the appropriate Rossby radius the basic state is irrotational, which
means that the right side of both these equations will be zero. This fact allows us to conclude that
Bernoulli’s equation (see below), which generally states the conservation of total energy along
streamlines, will in fact hold uniformly everywhere within the system under consideration. We
characterize the basic balance of pressure Pi, with kinetic and potential energy within each subdomain by using by these conservation relations at all levels (∀z) in the domain:
1
2
2
U1 +
P1
ρ1
+ gz = C1 ,
1
2
2
U2 +
P2
ρ2
+ gz = C 2 , with C1,C2 constant.
From which we derive a simple boundary condition at the interface between the domains, z = 0:
ρ1 ( 12 U 12 − C1 ) = ρ 2 ( 12 U 22 − C 2 ) .
(4.3)
We now introduce at time t=0 a perturbation to the interface between these domains. If it is
sufficiently small, but again in absence of viscosity, diffusion and other non-linear effects we have
chosen to ignore, Kelvin’s Circulation theorem tells us that the resulting system will still be
irrotational. In words, we require a fluid where net viscous forces vanish uniformly throughout
the domain (we have actually required an inviscid fluid), and where there is no stratification away
from the interface. In such a system, we state that any wave-like perturbation with a frequency
much greater than the local Coriolis frequency (and length scale sufficient to ignore dissipation,
diffusion, and surface tension) will have no relative vorticity.
To place these considerations in mathematical terms, we have defined a basic state and subjected it
to a certain class of perturbations, such that the resulting system can be characterized by two~
dimensional velocity potential functions φ 1,2 within each region:
~
φ1 = U 1 x + φ1 ,
~
φ2 = U 2 x + φ2 .
(4.4)
And further these functions will by definition obey the homogeneous Laplace equation:
~
∇ 2φ1 = 0 ,
~
∇ 2φ 2 = 0 .
We have in the process defined our perturbations in such a way that they are therefore also
characterized by potential functions, satisfying their own perturbation Laplace equations:
∇ 2φ1 = 0 ,
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∇ 2φ 2 = 0 .
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GFD-II 2007: Kelvin-Helmholtz Instabilities
(Note that a velocity potential function defines a scalar field, the components of whose gradient
defines two components of velocity. Where the flow is irrotational, as is the case here, the velocity
potential function completely defines fluid motions in the system, as: u≡∂φ/∂x, and v≡∂φ/∂y.)
We now subject the system to boundary conditions – the first of which is the rather weak
condition that this small-scale perturbation must be finite in vertical extent:
φ1 → 0 as z → ∞ ,
φ 2 → 0 as z → −∞ .
We next require a somewhat more stringent “kinematic” boundary condition, namely, continuity at
the interface. We in effect require that fluid, which is initially at either side of the interface
between the upper and lower layers, will remain at that interface as it develops:
∂φ1 dζ ∂ζ
∂ζ
∂ζ
=
=
+ (U 1 + u1 )
+ v1
∂z
∂t
∂x
dt
∂y
at z = ζ-, and
∂φ 2 dζ ∂ζ
∂ζ
∂ζ
=
=
+ (U 2 + u 2 )
+ v2
∂z
∂t
∂x
dt
∂y
at z = ζ+.
We here restrict the scale of our perturbation still further, by requiring that it be small enough for
non-linear terms in the perturbation velocities to be ignored – when we examine fluid parcel
motion at the basic interface at z=0. We thereby simplify the above conditions to:
∂φ1 ∂ζ
∂ζ
=
+ U1
∂z
∂t
∂x
at z = 0,
∂φ 2 ∂ζ
∂ζ
=
+U2
∂z
∂t
∂x
at z = 0.
(4.5)
We stated above a general constraint (4.3) on pressures in the basic flow state. Let us now look a
little closer at how pressure will be distributed across the perturbed interface. For note that in this
theoretical example, we have considered a fluid with discontinuous density and velocity interfaces.
But if our system is to be mathematically tractable and further is to have any relationship at all to
realistic situations, we must assume that pressure distribution across the system is continuous. We
are thus lead to state as a further boundary condition that the pressures on both sides of our
perturbed interface will smoothly approach the same limit.
To arrive at this condition, we first consider the balance between pressure and energy terms in the
perturbed system, above and below the interface. Thanks to our many careful assumptions, we
may state these balances in terms of the unsteady Bernoulli equation (Kundu 4.15):
~
∂φ1 1 ~ 2 ~
p
+ 2 (∇φ1 ) + 1 + gz = C1 ,
∂t
ρ1
~
∂φ 2 1 ~ 2 ~
p
+ 2 (∇φ 2 ) + 2 + gz = C 2 .
∂t
ρ2
(4.6)
From the above equation, it is obvious where our statement of the basic (pre-perturbation) pressure
balance (4.3) came from. However, we now strengthen that balance, requiring that the perturbation
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GFD-II 2007: Kelvin-Helmholtz Instabilities
p1 = ~
p2 at z=ζ. We must restate this condition in
pressures across the interface also be continuous: ~
terms of the potential functions and interface displacement. We can do this by restating the
unsteady Bernoulli equations in terms of our decomposition (4.4), and combining the two
equations into a single linear condition at z =ζ:
⎡
ρ1 ⎢C1 −
⎣
∂φ
∂φ1 1
⎡
⎤
⎤
− 2 (U 1 + u1 ) 2 + v12 + w12 − gζ ⎥ = ρ 2 ⎢C 2 − 2 − 12 (U 2 + u 2 ) 2 + v 22 + w22 − gζ ⎥
∂t
∂t
⎣
⎦
⎦
[
]
[
]
(4.7)
Again utilizing the scaling of our perturbation to restate this as a linear condition at the basic
interface, z=0, we then subtract our original basic state boundary condition 4.3 to simplify:
∂φ
∂φ
⎡ ∂φ
⎤
⎡ ∂φ1
⎤
+ U 1 1 + gζ ⎥ = ρ 2 ⎢ 2 + U 2 2 + gζ ⎥ ,
∂x
∂x
⎣ ∂t
⎦
⎣ ∂t
⎦
ρ1 ⎢
at z = 0.
(4.8)
Now we follow a procedure that will by now be very familiar to any student of large- and mesoscale geophysical dynamics. We assume that a vertically and horizontally decoupled linear
solution to this system may exist, allowing us to solve for all three of our variables above:
(ζ , φ1 , φ 2 ) = (ζ , φ1 , φ 2 )eik ( x −ct ) ,
) ) )
)
)
where k ∈ ℜ, but c = c r + ic i , and φ1 = Ae − kz , φ 2 = Be − kz (4.9)
(In the expressions for the lower and upper vertical perturbation modes, recall we linearized our
interface conditions to the level of the initial interface at z=0!) Our “kinematic” boundary
condition (4.5) of continuity at the interface then gives us immediately these solutions for A and B:
)
A = −i (U 1 − c)ζ
)
B = i (U 2 − c)ζ
(4.9a)
Substituting this with our presumed complete solutions into the unsteady Bernoulli equation, we
derive an expression relating our phase speed c to the horizontal wave number k:
kρ 2 (U 2 − c ) + kρ 1 (U 1 − c ) = g (ρ 2 − ρ 1 ) ,
2
2
(4.10)
Which we can easily solve for a direct expression of the (complex) phase speed, as:
⎛ U − U1 ⎞ ⎤
ρ U − ρ 1U 1 ⎡ g ρ 2 − ρ 1
⎟⎟ ⎥
c= 2 2
±⎢
− ρ 1 ρ 2 ⎜⎜ 2
ρ 2 + ρ1
⎢⎣ k ρ 2 + ρ 1
⎝ ρ 2 + ρ 1 ⎠ ⎥⎦
2
1/ 2
.
(4.11)
Clearly we find that c will have a non-zero imaginary part only in the case where the first term
within the square-root is less than the second term. For a stably stratified system (ρ2>ρ1), this can
only occur when U2 ≠ U1. From this it follows that our necessary criterion for mixing must be:
g (ρ22 - ρ12) < k ρ1ρ2 (U2 - U1)2.
Lew Gramer
(4.12)
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Compare this criterion (4.12) with that (4.0) argued purely from energy transfers above. If we
apply the Boussinesq approximation again (per Cushman-Roisin), they are in fact identical. The
sole apparent difference lies in the fact that our wavenumber k appears linearly on the right-hand
side of this inequality, in place of ∆H.
However, this difference is actually a chimera: we gave no criteria limiting ∆H in our energetics
argument above, and so now are free to choose ∆H = (1/k)! And this is actually an intriguing
choice, for it points to a fundamental relationship between the lateral and vertical scales in smallscale instabilities. And note that this in effect does then limit our vertical scale. For this relation
means that we impose a limit on our vertical scale ∆H based on the limits for an irrotational
system that we have chosen to impose on our horizontal scale. We will touch on this again later.
As shown in 4.11, both c and its complex conjugate c* are valid solutions to the vertical mode
problem. One of these is exponentially growing, the other exponentially decaying. Naturally, it is
the value of the conjugate pair with positive imaginary part that is our unstable (growing) solution.
Finally, recall that we have not excluded perturbations of any particular (small) scale a priori in
this theoretical analysis. Therefore, the simple presence of k above tells us that whenever our basic
condition of a non-zero shear is met – no matter how small the difference in velocities, or which
domain has the greater magnitude of mean velocity – that any perturbation of sufficiently short
length scale will always develop unstably, according to this criterion!
(As indicated at several other points in our discussion, though, this conclusion is actually not
really as absolute as it sounds, based on our assumptions. For in particular, we have chosen to
ignore viscosity, surface tension, density diffusion, and other terms that might potentially provide
a limiting lower bound on the scale of the perturbations that we are allowed to consider.)
Evolution into the continuous case
In our original energetics argument we made the implicit assumption that instability must
inevitably lead to mixing, and therefore homogenization of properties over some vertical subrange
of our domain. And in fact, in the normal mode wave analysis we have just done, we find (almost
incidentally) an implied vertical scale for that well-mixed sub-domain.
In fact, this evolution proves to be fundamental to all forms of Kelvin-Helmholtz instability. We
will see this for the continuous case on theoretical grounds from the Howard semi-circle theorem
below. And we will also be lead to find a more specific upper bound on mixing in a later section,
when we (briefly) discuss the dynamic transition from KHI, to full three-dimensional turbulence,
and so ultimately to efficient fluid mixing in boundary layers of the real atmosphere and ocean.
Thus we see that the discontinuous case we have discussed here is not merely unrealistic in itself.
But even where such discontinuities may exist, we will expect them to evolve quickly and
naturally into more realistic continuous gradients. So we are ultimately forced to consider a region
of shear in place of our discontinuity. From here forward then, we will examine a continuous
vertical shear profile in a horizontal mean flow. Further, we will allow our density profile to vary
in some continuous way as well, over a finite scale that is similar to that of our shear profile.
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Finally though, as we have hinted at in our discussion on boundaries above, we will still cling
throughout this paper to the idea of a domain that is theoretically infinite in horizontal extent.
5. Continuous profiles: the Taylor-Goldstein equation
As we have seen, where there is a velocity shear in the absence of the restoring force of buoyancy,
any perturbation at any wavelength we will allow ourselves to consider must lead inevitably to
instability. The kinetic energy available at the interface due to the difference in velocities will
always be sufficient to grow our perturbation. Where there is a discontinuous stratification on the
other hand, waves of sufficiently short wavelength are required to overcome the buoyancy force –
but again, where such waves can be of arbitrarily high wavenumber, instability always results.
Now we ask, how does this picture change with a continuous stratification profile – and in
particular when that profile co-exists with a continuous vertical mean flow shear in the same
region? This new situation is summarized by the following figure, adapted again from CushmanRoisin and Becker. Note that both the mean flows and the region of shear are drawn from the point
of view of an observer moving at the mean flow velocity of the system, i.e., |U+∞ – U-∞|. Also this
figure shows a linear vertical shear – though we will not need that assumption in our analysis:
Please note that for simplicity of notation, from here on subscripts denote partial derivatives.
The Euler equations (4.1) are again our starting point. However, we make several simplifying
assumptions. First following the Squires theorem, we allow ourselves to assume V ≡ v ≡ 0 without
loss of generality. Second we decompose our horizontal velocity, pressure and density into mean
and perturbation fields. Third we linearize the result of this Reynolds decomposition, to produce:
u t + wU z + Uu x = −
1
ρ0
px
wt + Uwx = −
1
ρ0
pz −
gρ
ρ0
Finally, we assume that our mean density distribution ρ is also invariant in x and y:
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ρ t + Uρ x + wρ z = 0
The result of these assumptions is that we may write our momentum- and density conservationequations in terms of a two-dimensional stream function ψ ∋: u = ψ z , w = −ψ x :
ψ zt − ψ xU z + ψ xzU = − ρ1 p x
(5.1a)
0
− ψ xt − ψ xxU = −
ρ t + Uρ x +
ρ0 N 2
g
1
ρ0
pz −
gρ
(5.1b)
ρ0
ψx =0
(5.1c)
Note the appearance of our old friend the Brunt-Väisälä frequency N, defined by N 2 ≡ −
g
ρ0
ρz .
We now apply the familiar separation of variables for all perturbed mean values, noting that we
)
)
)
allow our normal mode vertical amplitude functions ρ ( z ), p( z ),ψ ( z ) to take on complex values:
)
)
)
(U − c)ψ z − U zψ = − ρ10 p
(5.2a)
)
1 )
gρ
k (U − c)ψ = −
pz −
(5.2b)
2
)
ρ0
ρ0
2
) ρ N )
(U − c) ρ + 0 ψ = 0
g
(5.2c)
If we eliminate the pressure and density perturbation terms in the above equations, we derive a
single equation for the perturbation stream function, known as the Taylor-Goldstein equation:
⎤)
)
) ⎡ N2
(U − c) ψ zz − k 2ψ + ⎢
− U zz ⎥ψ = 0
⎦
⎣U − c
[
]
(5.3)
This is a simple second-order PDE, where the 0-th order coefficients represent, reading from left
to right: the kinetic energy of the current state of the wave perturbation, the stabilizing effect of
buoyancy on the wave, and the sapping of kinetic energy from the mean shear to the wave.
Here we follow Kundu (and Howard) in specifying finite boundary conditions for our vertical
structure function. But it is worth noting that these conditions are applied at whatever scale is
natural for our region of shear – and therefore, do not a priori establish any scale in themselves.
)
ψ
Lew Gramer
z =0
)
=ψ
z=H
=0
(5.4)
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Necessary condition: Richardson number criterion
The Taylor-Goldstein equation is an elegant expression of the relationship between restoring and
destabilizing forces in a parallel flow perturbation. But what exactly does it tell us about the
conditions for instability to develop in that flow? To see this more clearly, we will now embark on
some mathematical manipulation, in order to develop what is arguably the most famous and
satisfying result of small-scale stability theory – the Richardson number criterion.
Before beginning however, it is interesting to note in passing that some of the greatest minds of
the 20th Century were applied to the problem of sufficient conditions for small-scale stability (and
those for instability as well). Prior to the publication of a general, geophysically appropriate proof
of this criterion by Howard in 1961, no lesser lights than Ludwig Prandtl, G. I. Taylor and
Subrahmanyan Chandrasekhar (of black hole and supernova fame) all proposed critical stability
values ranging from 2 to 0.25, based on energetic, experimental, and other approaches. And it may
surprise the reader to hear that these seemingly conflicting results continue to have relevance in
recent years, for many sub-fields of physics – including geophysical flows, as evidenced by
important publications of Miles, Bayly and others as recently as the 1980s (e.g., Miles 1986).
To derive our criterion analytically as did Miles (and more generally after him, Howard), we will
begin with a simple conformal mapping of our perturbation stream function. Note that this new
motion potential function may still take on complex values:
φ=
)
ψ
U −c
This transformation in effect scales our field of time-varying motion, by the Doppler-shifted phase
speed of any perturbation. To see this more clearly, and to continue the proof, we find derivatives:
)
ψ z = (U − c)1 / 2 φ z +
)
φU z
2(U − c)1 / 2
ψ zz = (U − c) φ zz +
1/ 2
φ zU z + 12 φU zz
(U − c)1 / 2
1 φU z
−
4 (U − c) 3 / 2
2
With these results in hand, the Taylor-Goldstein equation (5.3) transforms to:
⎧
[(U − c )φ z ]z − ⎪⎨k 2 (U − c) + 12 U zz + 4
⎪⎩
1
2
U z − N 2 ⎫⎪
⎬φ = 0
(U − c) ⎪⎭
(5.5)
Naturally we also transform our boundary conditions for the T-GE, trivially as:
φ
z =0
=φ
z=H
=0
The terms in the differential equation (5.5) may be interpreted intuitively from left to right, as
follows: The first term, as a rate of energy transfer down the mean velocity gradient by shear in
the perturbation. The second and third terms, as contributions to that transfer through wave phase
propagation and curvature of the mean shear, respectively. And finally in the last coefficient of φ,
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we can see a balance of contributions to down-gradient transfer – between the positive effect of
the available mean-flow kinetic energy at each level, and the negative effect of buoyancy forces
(both upward and downward) at that same level.
Already in interpreting this final term, we glimpse a fundamental necessary condition for
instability! Instability may be viewed as a tendency for the mean-flow kinetic energy that is
available to displace particles and ultimately to mix a fluid, to increase with time. This requires
transfer of kinetic energy by a perturbation, and where the retarding force of buoyancy is sufficient
to disrupt that perturbation transfer of energy, stability is the inevitable result…
To confirm our intuitive interpretation above, we now use the “usual procedure” of stability
analysis, familiar to all GFD-II students by now, to derive a stability criterion analytically: we
multiply all terms by the conjugate of the amplitude, φ*, and integrate both terms over our vertical
domain. The resulting integral balance then becomes:
{
2
(
H
N 2 − 14 U z
2
∫0 (U − c) φ dz = ∫0 (U − c ) φ z
H
2
+ k2 φ
2
)−
1
2
2
}
U zz φ dz
(5.6)
As expected, the imaginary part of this integral balance can then be expressed as:
H
ci ∫
N 2 − 14 U z
0
U −c
2
2
H
(
)
φ dz = −ci ∫ φ z + k 2 φ dz
2
2
2
(5.7)
0
The integral on the right-hand side of equation (5.7) is always positive. However we can see that if
N2 > Uz2/4 at every level of our system, the integral on the left-hand side must also be positive, a
contradiction for non-zero ci. Thus if N2>Uz2/4, then our instability growth rate kci must inevitably
be zero. So happily, our intuition has proven out, and we can then state the Richardson-number
stability criterion for continuously stratified inviscid parallel flows as follows:
∀z , Ri ( z ) ≡
N2 1
> ⇒ complete stability of our system
U z2 4
(5.8)
In the literature, the name “Prandtl frequency” is sometimes used to refer to the term Uz2 above.
This is a wonderfully intuitive name, for in comparison with the stabilizing time scale of a density
gradient as described by the stratification (i.e., Brunt-Väisälä) frequency, this term describes the
destabilizing time scale that results from a kinetic energy gradient in our domain. The Richardson
number then, defines an appropriate scale for the relative values of these two frequencies, at which
we may always assume that stratification will dominate.
Why ¼? (Secondary instabilities, turbulence, and mixing)
As a final note, compare this Richardson criterion with our result for the discontinuous case (4.12).
The question arises as to why this result replaces a factor of 1 (i.e., direct inequality between shear
squared and squared density difference), with one of ¼. On this point, Miles, Howard, Thorpe,
Kundu (see references below) all appear to be silent. Intuition leads us to suspect that the
continuous case involves more degrees of freedom, leading to lower efficiency somehow in
converting mean kinetic energy into potential energy. How can this be clarified?
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The only answer I have found so far to this question in the literature – and it is a somewhat vague
one – is a comment by Cushman-Roisin (1994, p. 158): “The difference [between 1 and ¼] can be
explained by the difference in basic profile [discontinuous vs. continuous], and by the fact that the
analysis leading to (4.12) did not make provision for a consumption of kinetic energy by vertical
motions.” [Emphasis added.]
My interpretation of this is simple: instability must become three-dimensional for mixing to occur,
and in the process, the requisite kinetic energy increases by a factor of 22. And if this interpretation
is correct, it essentially means there is no conflict between our original mixing energy criterion
(4.0), and our Richardson number criterion (5.8) above. To put this point in simplistic terms, these
two criteria together tell us that energy required for mixing consists of one part wave-instability,
and at least three parts other motions – secondary instability, turbulent motion, and ultimately
momentum diffusion.
Finding sufficient conditions: empirical studies
Based on a separation of real and imaginary terms in the Taylor-Goldstein equation, we have just
found a fairly strong sufficient criterion for the stability of an inviscid, small-scale shear. But at
what values of the Richardson number for a region, can we safely assume that instability will exist?
To emphasize this point, note that the integral constraint embodied in the Richardson number
criterion only implies that a non-zero imaginary phase speed component cannot exist when Ri > ¼
everywhere. The converse of the condition is not always true. That is, when Ri(z) ≤ ¼ for some z
within our domain, we are not in general assured that there exists a wavelength at which any
perturbation may draw sufficient kinetic energy from the mean flow in order to grow.
What value of the Richardson number ensures instability? Such a value, where it exists, is called a
critical Richardson number, Ric. It is the strongest possible sufficient condition for stability, and
therefore the strongest possible necessary condition for instability of an inviscid system.
In fact, there is currently no general analytical solution to this problem. I was unable to find in the
literature I read, even an attempted statement or proof of any existence theorem for a critical
Richardson number. This appears to be a problem therefore, where the fundamental approach to
solving it must be inherently an empirical one. In that context, we acknowledge that any such
critical number found, will necessarily apply to a region, within a particular system.
However, saying that the critical Richardson number is primarily an experimental question, is not
the same as saying no analytical studies may be done. Kundu notes (1990, p 385) that critical
Richardson numbers can be derived for many theoretical profiles of stratification and shear – and
perhaps by implication for all such mathematically specified profiles, so long as they do not meet
the basic stability criterion. One example cited is a density profile that increases exponentially
with depth (very strong stratification), where a critical Richardson number above 0.2 does exist for
sufficiently sheared mean flow, e.g., Ric = 0.214 in the case of a jet with a profile like sech2(z):
Lew Gramer
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Figure 2: Simple numerical experiment by the author, indicating the critical Richardson number for a
mathematically defined profile where instability is known to result. Here u ∝ sech2(z), and ρ ∝ e(z / ∆H).
Theoretical studies like that above may be based on stratification and shear frequencies which
little resemble those found in observational meteorology and oceanography. However, it has been
observed in some real-world atmospheric and ocean flows that even when Ri(z) < ¼ at all levels of
a domain, the mean flow may still remain stable over relatively long time scales. This can often be
attributed to some of the circumstances we have specifically excluded so far, such as viscosity, the
presence upper or lower boundaries, or other factors that can stabilize small-scale flows. However,
there are observations where it is argued that these effects are not significant, and yet where the
Richardson criterion still fails to be a sufficient indicator of instability.
The first published paper citing observed values for critical Richardson numbers in oceanic flows
appears to be that of Woods (1968). However, Eriksen (1978) is recognized as having established
the Richardson number criterion as a reliable indicator of real fine-structure instabilities in the
ocean. Finally, statistical approaches to finding the appropriate critical Richardson number seem
to have been first considered by Bretherton (1969). Finding statistical moments appropriate to
given regions of the atmosphere or ocean based on surveys of real observational data, on the other
hand, appears to have been pioneered in the paper of Desaubies and Smith (1982).
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The reader interested in a broader perspective on observation and experiment into critical
Richardson numbers is encouraged to begin with some excellent review papers. The most recent
of these I could find is that by Riley and Lelong (2000); Fernando (1991) summarizes results from
several geophysically interesting numerical problems; while Miles (1986), now considered classic,
is brief but serves as a fine model for anyone in the conciseness of good scientific writing…
Finally, the paper by Thorpe (1971) might be called the seminal paper on experimental results and
their relationship to real instability and turbulence in the ocean. Below are two panels from that
paper, showing one of Thorpe’s experimental setups, and results that he obtained with it. Note in
particular the second panel, where each stage of the transition from stability, to KHI, to secondary
instability, to turbulence is documented both from above, and from the side:
The gist of these many references is that it is useful as a “rule of thumb”, to consider the minimum
Richardson number for a “characteristic vertical sampling” of a given geophysical or laboratory
flow system. (Often for realistic stratification profiles, this minimum will occur at the point where
the local vertical shear is greatest.) Ultimately, if this minimum sampled value is found to remain
below 0.25 for a sufficient sampling period, the researcher may expect shear instability to develop.
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Limits to growth: Howard’s semi-circle theorem
It is impossible in general, to directly relate the magnitude of a shear flow with the complex phase
speed of waves moving against that flow. At best, we hope for bounding relations.
The Howard semi-circle theorem in its simplest form first states a requirement on the speed of
propagation (cr) of a perturbation relative to the mean flow distribution, in order for instability to
be able to occur. The theorem then derives a theoretical absolute limit for the rate of amplitude
growth per unit wave number, ci, based on this phase speed cr and the shear of the background
flow. One interesting point about the theorem is that, subject to very few caveats, it applies to both
of the continuous and discontinuous vertical-shear cases we have described above, and also to
horizontally sheared barotropic fluid flows under rotation, briefly described in a following section!
The statement and the proof of this theorem for KHI is very similar to that already given for the
case of barotropic instability, in student seminar papers and textbooks for this class. The only
formal differences are, as we would expect: i) the replacement of beta (df/dy) with stratification
(density difference in the discontinuous case) in a restoring role; and ii) the replacement of
absolute vorticity gradient with simple velocity gradient, thereby simplifying both the inequality
relationship and the upper bound on growth. The result is summarized by this familiar graph:
A full statement and proof for our case is given in Kundu (1990, pp. 385-387). In briefest outline,
the proof begins with another non-linear coordinate transform of our perturbation stream function:
ϕ=
)
ψ
U −c
,
)
ψ z = (U − c)ϕ z + U z ϕ ,
)
ψ z = (U − c)ϕ zz + 2 ⋅ U z ϕ z + U zz ϕ .
It then uses the (by now, quotidian) trick of multiplying by the conjugate of the eigenfunction and
integrating over the vertical domain. The first result is a strong bounding relationship between the
mean flow distribution and unstable phase speeds: Umin < cr < Umax.
Second, Howard derives an upper bound for ci based on cr and the available kinetic energy of the
mean flow. The statement of this relationship appropriate to the case of KHI is:
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c i ≤ [12 (U max − U min )] − [c r − 12 (U max + U min )]
2
2
Finally, the weak upper bound on growth rate of the instability that trivially results is:
kci < (k/2) (Umax – Umin).
6. Meditations on the Taylor-Goldstein equation
The role of viscosity: the Reynolds number
Throughout this paper, we’ve chosen to ignore the effect of viscosity. At the small scales we
consider, this actually excludes the effects of molecular viscosity – as opposed to the so-called
“eddy viscosity” which dissipates larger-scale flows. Here we try to consider very briefly what
role molecular viscosity away from a boundary may play in small-scale instabilities. For
simplicity, we assume for this section only, a homogeneous fluid.
We begin by presuming the existence of a kinematic viscosity constant ν, which relates the
magnitude τ of the shear stress tensor at each level of our domain directly to the shear itself, by the
relation τ = ρνUz. If we allow this term to be considered in our equations of motion, intuition leads
us to expect that the energy lost from the mean flow in generating fluid particle deformation, will
not be available to feed any instabilities in the flow. In effect, this reasoning suggests, the
introduction of a viscosity term should have a stabilizing effect on the dynamics of a shear flow.
The magnitude of this stabilizing effect may be estimated using a well known non-dimensional
quantity, the molecular Reynolds number Re = ∆H⋅Umax/ν for some characteristic depth scale ∆H.
This compares the maximum available mean kinetic energy, with the presumed linear damping per
unit mass contributed by the production of stress – i.e., it compares inertial and viscous forces. We
consider the two-dimensional, irrotational momentum equations, but now using 1/ Re as a scale
factor for a damping term in the Laplacian of our velocity vector. We then assume a wave solution,
using separation of variables in a normal mode analysis, to derive (Kundu, pp. 387-391) a
governing equation for our vertical structure function φ(z), the Orr-Sommerfeld equation:
(U − c)(φ zz − k 2φ ) − U zz φ =
Subject as we might expect to:
(
1
φ zzzz − 2k 2φ zz + k 4φ
ikRe
)
(6.1)
φ=φz=0, at z=-∆H and z=+∆H.
Compare this to our Taylor-Goldstein equation (5.3) for the same vertical structure function. We
immediately note that the right-hand side here has replaced the simple -N2/(U-c) damping factor,
with a far more complex, fourth-order differential expression in the vertical structure. From
nothing other than the form of this right-hand term, we can quickly arrive at the three important
conclusions that significant with regard to viscosity’s role in instability:
1) Viscosity introduces a much higher order of complexity in the relationship between mean
shear and wave instability;
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GFD-II 2007: Kelvin-Helmholtz Instabilities
2) Any shear whose scale is comparable to that of molecular interactions, i.e., such that the
Reynolds number is not large, will significantly modify the tendency to instability; and,
3) One may not state a priori that viscosity always leads to the damping of instability.
Conclusion (3) above is perhaps the most important. For as the literature makes clear (e.g., Kundu
section 11.10), the effect of viscosity in shear flows is highly sensitive to many factors: Lateral or
vertical boundaries, small structural variations in the velocity shear, and details of the density
profile can all lead to significant changes in overall stability, when viscosity is significant. A fuller
treatment is beyond our scope, but interested readers are referred to Kundu, and to the many
papers in the literature, old and new, on this subject.
Comparison of barotropic and Kelvin-Helmholtz instability
The differential equation determining the horizontal structure of a perturbation in barotropic
rotational flow has been stated in class. In the form which will be most evocative to us, it is:
[
] [
]
)
)
)
(U − c) ψ yy − k 2ψ + β 2 − U yy ψ = 0
(6.2)
In the analysis of barotropic instability, this is labeled the Rayleigh equation, and its similarity to
the Taylor-Goldstein equation determining vertical structure of irrotational flows is striking. In
point of fact, they are actually the same equation with different antecedents! For Rayleigh is also a
general term used to describe a relation which specifies the transfer of energy from an available
mean kinetic-energy source, to a local potential-energy sink, possibly subject to a damping term.
I suppose this strong correspondence between instabilities in vertical/gravitational and
horizontal/rotational mean shear flows really should not surprise us. For Squire’s theorem (above)
seems to hint that two-dimensional instability is in fact a fundamental property of the continuous
equations of motion – whether they are stated in an accelerating or in an inertial coordinate frame.
The complexity that can result when the transfer of energy is allowed to occur in the opposite
direction, from mean potential-energy source to local kinetic-energy sink, leads us to the theory of
instabilities in baroclinic, rotational systems. The conditions that allow this “reversal” of energy
flows are very briefly touched on below. However, the student is lead to speculate philosophically
as follows – if barotropic and Kelvin-Helmholtz instabilities can be seen as two expressions of the
same principle, is there perhaps some still more fundamental principle, which in one relation can
be used to elicit all of the many classes of instability in fluid flow?
Dynamic similarity: lateral and vertical instability
We mentioned above the surprising (and fundamental) relationship between vertical stability in the
irrotational case, and larger-scale horizontal stability under rotation. Therefore, it should come as
no surprise that the Howard semi-circle theorem, as stated above, is applied equally well to any
instability that can be adequately considered in two spatial dimensions. In effect, Howard’s
theorem completes the definition of a dynamical similarity between the growth and propagation
rates of perturbations respectively derived from the Taylor-Goldstein and Rayleigh equations.
As we have seen in other presentations within this course, however, that lovely similarity does
break down with scale. This occurs as soon as we consider frequencies of stratification and shear
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(viz. the “Prandtl frequency” defined above) that are sufficiently small as to be comparable with
the planetary frequency. In effect, when the scales of shear and stratification become too large to
admit a decomposition of rotation and gravity, we must abandon our simple barotropic shallowwater or Euler equations, and fall back on quasi-geostrophic theory or similar approaches to
understand the development of instability in sheared flows.
However, this does not necessarily limit the power of these analyses – and in particular, the
application of Kelvin-Helmholtz theory to the real atmosphere and ocean. We now consider some
of these applications, forging (at least) an intuitive link from the “gravitationally dominated” scale
of the KHI, to the rotationally balanced scales of quasi-geostrophic or vorticity theories, and
ultimately leading to the rotationally dominated dynamics of the largest scales on our planet.
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7. Importance of Kelvin-Helmholtz instability
Surface gravity waves
Probably the most accessible example of Kelvin-Helmholtz instability in nature is the existence of
surface gravity waves. This is a situation where our discontinuous two-layer case, considered in
section 4, suddenly does not seem so realistic after all! At the air-sea interface, we find density
differences of 3 orders of magnitude, velocity shears of a smaller but significant scale acting on a
domain effectively infinite in lateral extent, and all within a vertical range of mere millimeters.
Applying an analysis like that we present above, it is obvious why sea waves develop. The only
factors that give us reason to pause in this conclusion are the role of surface tension, and also some
concern over the large difference in effective viscosities between the ocean surface and air. The
analysis of capillary waves – not detailed in this paper, but covered by Kundu (1990) at some
length – gives us a clear answer to the first concern. Namely, surface tension is an important, but
by no means dominant damper on instability at the smallest scales. And further, that wherever this
very small-scale instability does result in capillary surface waves, there is an obvious and natural
transition to the simpler Kelvin-Helmholtz form of instability over relatively short time scales.
The second concern is over viscosity, which as we have already stated has a very complex effect
on stability analysis. This concern is more difficult to address. The details are well beyond the
scope of this paper. And yet surface water waves do occur, and laboratory studies show that they
exhibit an etiology and constraints on growth strikingly similar to those we have defined above.
From this we surmise that surface gravity waves are a primary example of KHI – and one whose
details have important implications for global circulation, as we will now consider.
Role in transferring momentum from wind to current
The ultimate mechanism for momentum transfer from wind stress to ocean circulation is the finescale instability of the air-sea interface in the presence of strong density gradients – but where
there is even more powerful shear. And as was mentioned previously, the dominating dynamics of
these external gravity-wave instabilities, above wavelengths of about 7cm, is precisely the KelvinHelmholtz theory developed in section 5. KHI is thus seen to play a profoundly important role in
the coupling of the atmospheric and ocean general circulations, and so ultimately the climate
dynamics of the whole planetary system.
As I’ve tried to indicate, there is considerable complexity associated with analyzing the parameters
of this downward turbulent momentum flux. As a result, the process is often described in the
literature via a bulk formula, where the rate of wind stress on the ocean surface is related to wind
velocity by a constant multiplier, or simple piecewise-linear function CD, as: τ = ρCDU102. In
effect, this parameter relates the simple normalized wind velocity at 10-meters (U10), to the less
intuitive “frictional velocity” u*, which we introduce in the following section.
In practice however, as my readers should guess by now, this “simple factor” CD has proven to be
extraordinarily difficult to get right. And the basic structure of the world’s ocean circulations – and
so ultimately, their coupling with atmospheric dynamics as well – are clearly quite sensitive to this
parameterization. Major shifts in our picture of large-scale ocean circulation have resulted from
subtle changes in this parameter – viz., Bunker (1976), to Hellerman and Rosenstein (1983), and
even as recently significantly, as the important paper by Josey, Kent and Taylor in 2002.
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Ultimately, these shifts must affect our understanding of the basic dance of ocean and atmospheric
circulations and thermodynamics, which is the essence of climate dynamics. And clearly, from the
arguments we’ve presented above, the specifics of this parameterization in the real world must
depend critically on our understanding of Kelvin-Helmholtz and related instabilities. So as we
suggested in the introduction, these complexities at the smallest scales have a profound impact.
Homogenization – ABL and oceanic ML
We have just gained a simple intuitive understanding of the direct impact of KHI on the winddriven dynamics of the ocean. But how does it affect the air-sea coupling in the other direction? In
other words, does Kelvin-Helmholtz instability offer us any help in understanding the ocean’s
affect on the atmosphere? The answer to this question of course, is yes.
To adequately answer this would of course require a detailed review of the thermodynamics of airsea temperature and moisture exchange. And what is event more daunting, it would ultimately
demand a review of the theory of other air-sea exchanges as well, viz. the CO2 transfer-velocity
problem and its impact on climate dynamics! This all is clearly far, far beyond our scope here.
However it may suffice to say here, that many recent and classic papers suggest these exchange
problems all relate in a fundamental way to the dynamics of the two mixed layers that lie at the
air-sea interface: the oceanic mixed layer (ML), and the atmospheric boundary layer (ABL). To
understand this, recall that two of the primary drivers of upward air-sea coupling are sea surface
temperature (SST) and sea-surface evaporation.
These phenomena occur over the few millimeters of the air-sea interfacial surface, or “skin”, and
at the levels jus below – where they are significantly affected by the wave dynamics we hinted at
in the previous section. But in fact, if these exchanges are to continue over time scales sufficient to
affect climate, we know that the energy required cannot come simply from this upper ocean layer.
Rather, the source of significant energy transfer from the ocean to the atmosphere must ultimately
lie below, in the mixed layer – and in the processes that can transport that energy upward.
So we see that coupled climate dynamics must be sensitive to processes in the oceanic mixed layer.
Therefore I will now try, in fewest possible words, to help the reader see the connection between
Kelvin-Helmholtz instability, and the depth and structure of the oceanic mixed layer in particular.
Monin-Obukhov depth scale
We follow Cushman-Roisin, and recall the relationship between shear, stratification and mixing
( ρ − ρ1 )
g ⋅ ∆H ≤ (U 2 − U 1 ) 2 .
depth embodied in inequality (4.0): 2
ρ
We note that (U1 – U2)2 may be interpreted as a square of the local variation in horizontal velocity,
over a region defined by our mixing interval ∆H. This local variation, when considered in a threedimensional context, is frequently described as the frictional velocity u*. Similarly, the density
difference (ρ2 – ρ1) can be seen in a suitably scaled regime as a local density fluctuation, e.g., one
due to three-dimensional turbulent processes over some scale just larger than our depth interval.
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Under the proper circumstances then, these two multiplied together constitute a local vertical
momentum flux (and note where instability is present, this is an upward flux). In the Reynolds
decomposition of the original Euler equations then, this term must be expressed as a time-mean
over the product of two time fluctuations, as follows: w′ρ ′ .
As we indicated in section, the mixing due to Kelvin-Helmholtz instability has a natural depth
scale. If we recast this depth scale in terms of the frictional velocity and the Reynolds terms
described above, the result is something referred to as the Monin-Obukhov mixing depth:
ρ u *3
.
L ≡ ∆H =
κg w′ρ ′
(Kundu, 1990, pp. 460-465)
In the above formula κ is a thermodynamic parameter related to our old friend, von Karman’s
constant k. Note that this scale defines a vertical depth that we may expect mixing from an upper
interface to ultimately reach. However, as we’ve indicated in previous sections, it also defines a
corresponding horizontal length that perturbations must achieve in order to effect this mixing.
As a concluding point, we ask what determines this parameter κ in the real ocean? We cannot
discuss this question in detail here, as again it would lie well, well beyond our very limited scope.
However, the reader should not be surprised by now that the answer to this question relies – at
least in part – on the conditions that lead to Kelvin-Helmholtz instability. Ultimately then, we may
conclude that the value of the critical Richardson number, for real vertical sections of the ocean’s
upper layers, in fact determines (to some degree at least), the energy which can be made available
for the coupling between ocean and atmospheric dynamics…
The “k parameterization” problem
As a final, painfully brief note, we must mention the role of Kelvin-Helmholtz instability in the
modeling of general ocean circulation – again making the point that an understanding of the very
smallest-scale dynamics must ultimately modify our view of the very largest.
As we know, ocean (and presumably atmospheric) modeling relies on the relatively simplicity of
the Navier-Stokes equations – that is, simplicity in their expression, albeit not their solution! And
to make the calculation of numerical solutions to these equations computationally feasible, finitedifference modeling relies on our ability to limit grid-scales. That is, a model must assume that all
the “important” dynamics occur at scales that are large relative to the size of the squares in its grid.
Wherever this is not the case, some simple expression must be found for these “sub-grid scale”
dynamics. For this, modelers rely on parameterizations of a downward energy cascade, wherein
energy in motions at larger scales breaks down, or is absorbed, into energies of motion at ever
smaller and smaller scales, ending in dissipation. We can summarize this simplified view using the
familiar picture below, courtesy of Cushman-Roisin and Beckers (2007).
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And yet what governs the rate of energy transfer or “dissipation”, from the larger scales to the
smaller. In fact, as we hinted at in the introduction, this is in fact the result of instabilities. That is,
perturbations arising at the smallest scales must presumably result in an upward cascade of
instabilities from one scale to the next. This is the only logical conclusion: closure requires us to
understand smaller scales first. And subject to a few caveats, we ask, what is the smallest scale
instability that we have so far encountered in our studies of the physical world? The answer as the
reader will have long-since guessed, is in fact our friend, Kelvin-Helmholtz…
Conclusion
This paper gives a basic mathematical treatment of small-scale perturbations in sheared mean
flows, and analyzes a particular kind of mean-flow instability that results from these perturbations.
I have also tried to give some insight into the significance of this Kelvin-Helmholtz instability,
and its analysis, in our larger view of geophysics. This latter part has been almost purely intuitive
in character. And yet ultimately, I think we can find some solid footing for all of these intuitions in
the real world, if we only consider just how ubiquitous instances of Kelvin-Helmholtz instability
really do seem to be! Therefore, I conclude this treatment of KHI with a few simple images of this
phenomenon actually occurring. These images are drawn both from nature and from the rich
literature of laboratory experiments in fields that rely on KHI analysis, and are presented purely
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GFD-II 2007: Kelvin-Helmholtz Instabilities
for the reader’s enjoyment. For many more such images the reader is referred to wonderful online
galleries of the Earth-science Picture of the Day (EPOD), and of the journal Physics of Fluids:
http://epod.usra.edu/
http://pof.aip.org/pof/gallery/
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Lew Gramer
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Acknowledgements
I am particularly indebted to Professor Zhang and to Emily Riley, for their insightful comments
and content-recommendations. Further, my most profound thanks are due to Dr. Sang-Ki Lee,
both for his stimulating discussions with me on ocean modeling and other topics, and for his
original recommendation of Kundu’s book as my starting point for researching this paper.
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