Table Of ContentUnder consideration for publication in J. Fluid Mech. 1
Energy and potential enstrophy flux
constraints in the two-layer
quasi-geostrophic model
By Eleftherios Gkioulekas
2
University of Texas-Pan American, Department of Mathematics, Edinburg TX, 78539-2999,
1
USA
0
2
(Received 5 June 2012)
n
u We investigate an inequality constraining the energy and potential enstrophy flux in the
J two-layer quasi-geostrophic model. This flux inequality is unconditionally satisfied for
1 the case of two-dimensional Navier-Stokes turbulence. However, it is not obvious that it
remainsvalidunderthemulti-layerquasi-geostrophicmodel.Thephysicalsignificanceof
] this inequality is that it decides whether any given model can reproduce the Nastrom-
D
Gage spectrum of the atmosphere, at least in terms of the total energy spectrum. We
C
derivethegeneralformoftheenergyandpotentialenstrophydissipationratespectrafor
.
n a generalized multi-layer model. We then specialize these results for the case of the two-
i layer quasi-geostrophic model under dissipation configurations in which the dissipation
l
n terms for each layer are dependent only on the streamfunction or potential vorticity of
[
that layer. We derive sufficient conditions for satisfying the flux inequality and discuss
2 the possibility of violating it under different conditions.
v
7
6
5
0 1. Introduction
.
1 It is now well-known that in two-dimensional Navier-Stokes turbulence, most of the
0
energy tends to go towards larges scales and most of the enstrophy tends to go towards
2
small scales. This was initially proposed by Fjørtøft (1953) via his triad interactions
1
: argument. Later, Kraichnan (1967), Leith (1968), and Batchelor (1969) introduced the
v
theory that the energy forms an upscale inverse energy cascade with energy spectrum
i
X scaling as k−5/3, and that the enstrophy forms a downscale enstrophy cascade with k−3
r scaling. Kraichnan (1967) argued, differently from Fjørtøft (1953), that the direction of
a
the two cascades can be justified via a thermodynamic argument in which we introduce,
without proof, the assumption that the energy and enstrophy fluxes should tend to
revert the energy spectrum from a cascade configuration to the absolute equilibrium
configuration.Theexistenceofforcinganddissipationarreststhistendency,thuskeeping
the system locked in a steady-state forced-dissipative configuration away from absolute
equilibrium.
Merilees&Warn(1975)identifiedaseriouserrorwiththeoriginal Fjørtøftargument:
Fjørtøft (1953) claimed that in every triad interaction group, more energy is transferred
upscale than downscale. However, a more rigorous analysis shows that there exist triad
interaction groups in which more energy is sent downscale than upscale, and it is not
obvious, without additional considerations, as to which group is dominant. This was
explainedindetailalsobyGkioulekas&Tung(2006).Asidefromthismatter,thefunda-
mental problem that underlies every other proof that utilizes only the twin conservation
lawsofenstrophyandenergy,isthatanadditionalassumptionneedstobeintroducedto
2 Eleftherios Gkioulekas
overcomethesymmetryoftheEulerequationsundertimereversal.Typicalassumptions,
such as the tendency of the energy spectrum to revert to absolute equilibrium, or the
tendency of an energy peak to spread, typify ad hoc constraints imposed implicitly on
the initial conditions that are needed to break the time reversal symmetry. All of these
proofs have been reviewed in detail by Gkioulekas & Tung (2007b). More importantly,
Gkioulekas&Tung(2007b)havecounterproposedaverysimpleandmathematicallyrig-
orousproofthatavoidstheneedforanyadhocassumptionsbyconsideringthecombined
effect of the Navier-Stokes nonlinearity and the dissipation terms. The only assumption
used by this proof is that the forcing spectrum is restricted to a finite interval [k ,k ]
1 2
of wavenumbers. Later, Farazmand (2010) and Farazmand, Kevlahan & Protas (2011)
showed that even that assumption can be relaxed to some extent, although not entirely
eliminated.
The essence of the argument is to show that for every wavenumber k not in the forc-
ing range, the energy flux Π (k) and the enstrophy flux Π (k) satisfy the inequality
E G
k2Π (k) − Π (k) ≤ 0. Here, Π (k) represents the amount of energy per unit vol-
E G E
ume transferred from the wavenumbers in the (0,k) interval to the wavenumbers in the
(k,+∞) interval, and Π (k) is defined similarly for the enstrophy. From this inequality
G
we then derive the following integral constraints for Π (k) and Π (k):
E G
(cid:90) k
qΠ (q)dq ≤0, ∀k ∈(k ,+∞), (1.1)
E 2
0
(cid:90) +∞
q−3Π (q)dq ≥0, ∀k ∈(0,k ). (1.2)
G 1
k
As explained by Gkioulekas & Tung (2007b), these constraints imply a predominantly
upscale transfer of energy and a predominantly downscale transfer of enstrophy. The
original flux inequality k2Π (k)−Π (k)<0 itself can also be directly interpreted as a
E G
tight constraint on the downscale energy flux.
Thequestionthatconcernsusinthepresentpaperiswhetherthisfluxinequalitycon-
tinuestoholdformulti-layerquasi-geostrophicmodels.Quasi-geostrophicturbulence,as
shownrecentlybyLindborg(2007),isareasonableapproximationforthecascadedynam-
ics of atmospheric turbulence at length-scales larger than 100km, in order of magnitude.
Underlying the model are the assumptions of small thickness and fast rotation. Towards
large scales, the atmospheric thickness becomes relatively small and the Coriolis effect
of rotation becomes increasingly noticeable. In multi-layer quasi-geostrophic models, the
vertical dimension is discretized into a finite number of layers. The simplest layer model
isoneinwhichweonlyhavetwolayers.Thistwo-layerquasi-geostrophicmodelwassimu-
latedbyTung&Orlando(2003),whoclaimedthatitcanreproducetheenergyspectrum
of the atmosphere, consistently with the measurement of Gage & Nastrom (1986). Tung
&Orlando(2003)intendedtocorroboratetheirtheorythattheNastrom-Gagespectrum
consists of a downscale potential enstrophy cascade coexisting with a downscale energy
cascadeoverthesamerangeofscales.Gkioulekas&Tung(2005a,b)clarifiedthetheoret-
ical mechanism that allows cascades to coexist and predicted that each cascade provides
a contributing term to the energy spectrum and that both contributions are linearly su-
perposed with one another. This linear superposition hypothesis and the Tung-Orlando
theory of coexisting cascades have been both corroborated by the recent measurements
andanalysisbyTerasaki,Tanaka&Zagar(2011).Arecentprimitiveequationsnumerical
simulation by Vallgren, Deusebio & Lindborg (2011) also seems to support the notion
of a downscale energy cascade coexisting with a downscale potential enstrophy cascade,
and a more detailed discussion of these matters has been given by Gkioulekas (2012).
the two-layer quasi-geostrophic model 3
More specifically, the Nastrom Gage spectrum of the atmosphere consists of a k−3
slopeintheenergyspectrumthattransitionstok−5/3 atalengthscaleofapproximately
1000kmto700km,neartheRossbydeformationwavenumberk .Accordingtothetheory
R
ofTung&Orlando(2003)andGkioulekas&Tung(2005a,b),theNastrom-Gagespectrum
arises from coexisting downscale cascades of potential enstrophy and energy, over the
samerangeofscales,eachcascadegivesacontributiontotheenergyspectrum,andboth
contributions are combined linearly. As a result, the k−3 contribution of the downscale
potential enstrophy cascade is dominant for small wave numbers k, only to be overtaken
bythek−5/3 contributionofthedownscaleenergycascadenearatransitionwavenumber
k . For the k−5/3 contribution of the downscale energy cascade to become dominant
t
at wavenumbers k ∼ k beyond a transition wavenumber k , it is necessary for the
t t
flux inequality to break down near k and reverse direction for wavenumbers k > k .
t t
Consequently, the direction of the flux inequality decides whether various models can
reproduce the Nastrom-Gage spectrum, at least in terms of the total energy.
For the case of two-dimensional turbulence, as was first noted by Gkioulekas & Tung
(2005a,b), the unconditional validity of the flux inequality constitutes a mathemati-
callyrigorousproofthattwo-dimensionalturbulencecannotreproducetheNastrom-Gage
spectrumunderanycircumstances,aslongasthedissipationtermsaregivenintermsof
integerorfractionalpowersofLaplacianoperators.Forthecaseoftherealatmosphere,it
isfairlyobviousthatthefluxinequalityisviolated,providedthatweagreewiththeTung-
Orlandotheory(Tung&Orlando2003)thattheNastrom-Gagespectrumconsistsoftwo
coexisting downscale cascades of potential enstrophy and energy. If the flux inequality
can be also violated by quasi-geostrophic models, then these models can reproduce the
Nastrom-Gage energy spectrum, at least in terms of the total energy spectrum E(k), if
not in terms of a correct distribution of energy between kinetic energy and potential en-
ergy. If, on the other hand, the flux inequality continues to hold, then quasi-geostrophic
models cannot reproduce the Nastrom-Gage spectrum either. The question that we ulti-
mately want to investigate is whether the quasi-geostrophic models have enough physics
to enable a break-down of the flux inequality, and what configuration of the dissipation
terms is needed to accomplish that.
Gkioulekas & Tung (2007a) first noticed that the only mechanism that can cause
a flux inequality violation is an asymmetry in the dissipation operators for different
layers. Unfortunately, without more information about the phenomenology of the two-
layer quasi-geostrophic model, and multi-layer quasi-geostrophic models in general, it is
verydifficulttoderiveanysufficientconditionsfortheneededfluxinequalitybreakdown.
Ontheotherhand,itispossibletoderiverigoroussufficientconditionsforsatisfyingthe
flux inequality when the asymmetry is restricted by an upper bound, without assuming
anyknowledgewhatsoeverofthephenomenologyofanyofthespectrainvolved.Thegoal
of the present paper is to address this non-controversial aspect of the larger problem, by
derivingaseriesofsuchresults,forthecaseofweakdissipationasymmetry,andindoing
so, to also set up the necessary mathematical infrastructure for a further exploration of
the flux inequality. Unlike the case of two-dimensional turbulence, where the derivation
of the flux inequality is easy, it should be clear from the present manuscript that the
flux inequality in the quasi-geostrophic model is both nontrivial and sensitive to the
configuration of the dissipation terms across layers.
This paper is organized as follows. In section 2, we introduce the generalized multi-
layer model and its conservation laws of energy and layer-by-layer potential enstrophy.
In section 3, we introduce the budget equations for the energy and potential enstrophy
spectra, derive the required conditions for the energy spectrum definition to remain
positive definite, and derive general expressions for the energy and potential enstrophy
4 Eleftherios Gkioulekas
dissipation rate spectra. Simplifications of these general expressions are given for the
specialcasesofstreamfunctiondissipationandpotentialvorticitydissipation.Fromthese
results, in section 4 we derive sufficient conditions for satisfying the flux inequality for
the special case of the two-layer quasi-geostrophic model under the two special cases
of streamfunction dissipation and potential vorticity dissipation. Because the details of
the calculation are tedious, the main results are summarized in section 5. Conclusions
and general discussion are given in section 6. Technical matters are taken up in the
appendices.
2. The generalized multilayer model and conservation laws
Following Gkioulekas (2012), we write the governing equations for the generalized
multi-layer model in matrix form:
∂q
α +J(ψ ,q )=d +f , (2.1)
∂t α α α α
(cid:88)
d = D ψ . (2.2)
α αβ β
β
Hereψ representsthestreamfunctionattheα-layer,q representsthepotentialvorticity
α α
at the α-layer, D is a linear operator encapsulating the dissipation terms, and f is
αβ α
the forcing term acting on the α-layer. The index α takes the values α = 1,2,...,n
representingthelayernumber,foramodelinvolvingnlayers.Sumsoverindices,suchas
in the sum over the index β in the dissipation terms above, are assumed to run over all
layers1,2,...,n,unlessweindicateotherwise.Itisalsoassumedthatthestreamfunction
ψ and the potential vorticity q are related via a linear operator L according to:
α α αβ
(cid:88)
q (x,t)= L ψ (x,t). (2.3)
α αβ α
β
Theaboveequationsencompassboththetwo-layerquasi-geostrophicmodelandthemul-
tilayer quasi-geostrophic model, on the assumption that we neglect the β-effect, arising
fromthelatitudinaldependenceoftheCoriolispseudoforce.Thisisareasonableassump-
tion for Earth, especially if we restrict our interest to a thin strip of the Earth’s surface,
oriented parallel to the equator. Baroclinicity instability is accounted for by the forcing
term f , and implicit in the entire argument is the assumption that it forces the system
α
at large scales only. This assumption, originally proposed by Salmon (1978, 1980), is the
onlyphysicalassumptionimplicitinthetheoreticalframeworkofthefluxinequality,and
it has been corroborated by Welch & Tung (1998) and Tung & Orlando (2003).
Forthesakeofsimplifyingouranalysis,weassumethatallfieldsaredefinedinaninfi-
nite two-dimensional domain. Then we can write the Fourier expansions for the stream-
function ψ and the potential vorticity q as follows:
(cid:90)
ψ (x,t)= ψˆ (k,t)exp(ik·x)dk, (2.4)
α α
R2
(cid:90)
q (x,t)= qˆ (k,t)exp(ik·x)dk. (2.5)
α α
R2
We assume that the operator L is diagonal in Fourier space. This means that the
αβ
relation between the streamfunction and the potential vorticity, in Fourier space, reads:
qˆ (k,t)=(cid:88)L ((cid:107)k(cid:107))ψˆ (k,t). (2.6)
α αβ α
β
the two-layer quasi-geostrophic model 5
We also assume that L is symmetric with L = L . This implies that L (k) =
αβ αβ βα αβ
L (k)forallwavenumbersk.Forann-layerquasi-geostrophicmodel,theL (k)matrix
βα αβ
becomes a tridiagonal matrix. For the special case of the two-layer quasi-geostrophic
model, the matrix L (k) is given by
αβ
(cid:20)−k2−k2/2 +k2/2 (cid:21)
L (k)= R R , (2.7)
αβ +k2/2 −k2−k2/2
R R
with k the Rossby wavenumber. For quasi-geostrophic models, the matrix L (k) is
R αβ
non-singular for all wavenumbers k > 0, due to being diagonally dominant, and we
assume that to be the case in our abstract formulation given above. Consequently, there
is an inverse matrix L−1(k) which defines the inverse operator L−1. To accommodate
αβ αβ
a possible singularity at k = 0 we assume that at wavenumber k = 0, in Fourier space,
thecorrespondingfieldcomponentis0forallfields.Thisisequivalenttosubtractingthe
mean field and considering only the field fluctuation around the mean.
Likewise, we assume that the dissipation operation D is also diagonal in Fourier
αβ
space and that the Fourier expansion of the dissipation term D ψ reads:
αβ β
(cid:90)
(D ψ )(x,t)= D ((cid:107)k(cid:107))ψˆ (k,t)exp(ik·x)dk. (2.8)
αβ β αβ β
R2
In general, the dissipation operator is a matrix operator that is allowed to entangle
multiple layers together. Realistically, this occurs when one follows Salmon (1978, 1980)
in the definition of the Ekman term, which should appear only on the bottom layer
but entangles both layers together. Previously, Tung & Orlando (2003), Gkioulekas &
Tung (2007a), and Gkioulekas (2012), restricted their attention to diagonal dissipation
only where D (k) = 0 for α (cid:54)= β. In this case, the spectrum D (k) of the dissipation
αβ αβ
operator reads D (k)=δ D (k) with δ given by
αβ αβ β αβ
(cid:26)
1, if α=β
δ = . (2.9)
αβ 0, if α(cid:54)=β
We will now show that the generalized layer model, in the absence of dissipation,
conserves the total energy E and the total potential enstrophy G under very general
conditions on the operator L , For any arbitrary scalar field f(x,y) we write the cor-
αβ
responding volume integral using the following notation:
(cid:90)(cid:90)
(cid:104)(cid:104)f(cid:105)(cid:105)= f(x,y)dxdy. (2.10)
R2
We define the total energy E over all layers, and the layer-by-layer total potential en-
strophy G for layer α, as E = −(cid:80) (cid:104)(cid:104)ψ q (cid:105)(cid:105) and G = (cid:104)(cid:104)q2(cid:105)(cid:105). The purpose of the
α α α α α α
minus sign in our definition of the total energy E is to maintain consistency with the
notation and sign conventions used by Gkioulekas (2012). Specifically, we will show that
the potential enstrophy is conserved on a layer-by-layer basis unconditionally regardless
of the details of the operator L . Conservation of the total energy E, over all layers,
αβ
on the other hand, requires that the operator L be symmetric and self-adjoint. By
αβ
symmetric we mean that the operator satisfies L = L . To define the self-adjoint
αβ βα
property, consider two arbitrary two-dimensional scalar fields f(x,y) and g(x,y). We re-
quire that every component of the operator L must satisfy (cid:104)(cid:104)f(L g)(cid:105)(cid:105)=(cid:104)(cid:104)(L f)g(cid:105)(cid:105)
αβ αβ αβ
for any two fields f(x,y) and g(x,y). This self-adjoint property, so defined, follows as an
immediate consequence of our previous assumption that the operator L is diagonal in
αβ
Fourier space. In the proof given below, however, there is no need to use the stronger
assumption of diagonality.
6 Eleftherios Gkioulekas
TheproofisbasedonthefollowingpropertiesofthenonlinearJacobianterm.Ifa(x,y)
andb(x,y)aretwo-dimensionalscalarfieldsthatsatisfyahomogeneous(DirichletorNeu-
mann) boundary condition, then we can show that (cid:104)(cid:104)J(a,b)(cid:105)(cid:105) = 0, using integration by
parts. Then, we note that, as an immediate consequence of the product rule of differen-
tiation, given three two-dimensional scalar fields a(x,y),b(x,y), and c(x,y) we have
(cid:104)(cid:104)J(ab,c)(cid:105)(cid:105)=(cid:104)(cid:104)aJ(b,c)(cid:105)(cid:105)+(cid:104)(cid:104)bJ(a,c)(cid:105)(cid:105)=0, (2.11)
from which we obtain the identity
(cid:104)(cid:104)aJ(b,c)(cid:105)(cid:105)=(cid:104)(cid:104)bJ(c,a)(cid:105)(cid:105)=(cid:104)(cid:104)cJ(a,b)(cid:105)(cid:105). (2.12)
Now, let us go ahead and drop the dissipation and forcing terms and write the time-
derivative of the potential vorticity q as q˙ =−J(ψ ,q ). Then, the time derivative of
α α α α
the streamfunction ψ reads:
α
ψ˙ =(cid:88)L−1q˙ =−(cid:88)L−1J(ψ ,q ). (2.13)
α αβ β αβ β β
β β
Differentiating the total potential enstrophy G for the α layer with respect to time
α
and employing the identity given by Eq. (2.12) immediately gives:
G˙ =2(cid:104)(cid:104)q q˙ (cid:105)(cid:105)=−2(cid:104)(cid:104)q J(ψ ,q )(cid:105)(cid:105)=−2(cid:104)(cid:104)ψ J(q ,q )(cid:105)(cid:105)=0. (2.14)
α α α α α α α α α
Here, we note that from the definition of the Jacobian J(q ,q ) = 0. This establishes
α α
the layer-by-layer conservation law of potential enstrophy, unconditionally, as claimed.
To show the energy conservation law, we differentiate the total energy E with respect to
time and obtain:
E˙ =−(d/dt)(cid:88)(cid:104)(cid:104)ψ q (cid:105)(cid:105)=−(cid:88)(cid:104)(cid:104)ψ˙ q (cid:105)(cid:105)−(cid:88)(cid:104)(cid:104)ψ q˙ (cid:105)(cid:105) (2.15)
α α α α α α
α α α
(cid:88) (cid:88)
= (cid:104)(cid:104)q L−1J(ψ ,q )(cid:105)(cid:105)+ (cid:104)(cid:104)ψ J(ψ ,q )(cid:105)(cid:105) (2.16)
α αβ β β α α α
αβ α
(cid:88) (cid:88)
= (cid:104)(cid:104)J(ψ ,q )L−1q (cid:105)(cid:105)+ (cid:104)(cid:104)q J(ψ ,ψ )(cid:105)(cid:105) (2.17)
β β αβ α α α α
αβ α
(cid:88) (cid:88)
= (cid:104)(cid:104)J(ψ ,q )L−1q (cid:105)(cid:105)= (cid:104)(cid:104)J(ψ ,q )ψ (cid:105)(cid:105) (2.18)
β β βα α β β β
αβ β
(cid:88)
= (cid:104)(cid:104)J(ψ ,ψ )q (cid:105)(cid:105)=0. (2.19)
β β β
β
Note that the self-adjoint property is applied at step Eq. (2.17), and the symmetric
property is applied at Eq. (2.18). This concludes the proof.
3. Spectra and budget equations
Following Frisch (1995) and Gkioulekas (2012), we define spectra for the energy and
potential enstrophy using the bracket notation, which is defined as follows. Consider,
in general, two arbitrary two-dimensional scalar fields a(x) and b(x). Let a<k(x) and
b<k(x) be the fields obtained from a(x) and b(x) by setting to zero, in Fourier space,
the components corresponding to wavenumbers whose norm is greater than k. Formally,
a<k(x) is defined as
(cid:90) (cid:90) H(k−(cid:107)k (cid:107))
a<k(x)= dx dk 0 exp(ik ·(x−x ))a(x ), (3.1)
0 0 4π2 0 0 0
R2 R2
the two-layer quasi-geostrophic model 7
with H(x) the Heaviside function, defined as the integral of a delta function:
(cid:90) x 1, if if x∈(0,+∞)
H(x)= δ(τ)dτ = 1/2, if if x=0 . (3.2)
0 0, if if x∈(−∞,0)
Obviously, b<k(x) is defined similarly. We now use the two filtered fields a<k(x) and
b<k(x) to define the bracket (cid:104)a,b(cid:105) as:
k
(cid:104)a,b(cid:105) = d (cid:90) dx (cid:10)a<k(x)b<k(x)(cid:11) (3.3)
k dk
R2
1(cid:90) (cid:68) (cid:69)
= dΩ(A) [aˆ∗(kAe)ˆb(kAe)+aˆ(kAe)ˆb∗(kAe)] . (3.4)
2
A∈SO(2)
Here, aˆ(k) and ˆb(k) are the Fourier transforms of a(x) and b(x), SO(2) is the set of all
non-reflecting rotation matrices in two dimensions, dΩ(A) is the measure of a spherical
integral,eisatwo-dimensionalunitvector,and(cid:104)·(cid:105)representstakinganensembleaverage.
The star superscript represents taking the complex conjugate. Note that Eq. (3.3) is the
definitionofthebracket,andEq.(3.4)followsfromEq.(3.3)asaconsequence.Asnoted
by Gkioulekas (2012), the bracket satisfies the following properties:
(cid:104)a,b(cid:105) =(cid:104)b,a(cid:105) , (3.5)
k k
(cid:104)a,b+c(cid:105) =(cid:104)a,b(cid:105) +(cid:104)a,c(cid:105) , (3.6)
k k k
(cid:104)a+b,c(cid:105) =(cid:104)a,c(cid:105) +(cid:104)b,c(cid:105) . (3.7)
k k k
Moreover, every (αβ)-component of the operator L is self-adjoint with respect to the
αβ
bracket, which gives
(cid:104)L a,b(cid:105) =(cid:104)a,L b(cid:105) =L (k)(cid:104)a,b(cid:105) , (3.8)
αβ k αβ k αβ k
andthesamepropertyisalsosatisfiedbyeverycomponentoftheinverseoperatorL−1:
αβ
(cid:68) (cid:69) (cid:68) (cid:69)
L−1a,b = a,L−1b =L−1(k)(cid:104)a,b(cid:105) . (3.9)
αβ αβ αβ k
k k
(cid:80)
Usingthebracket,wedefinetheenergyspectrumE(k)=− (cid:104)ψ ,q (cid:105) ,andwealso
α α α k
define the layer-by-layer potential enstrophy spectrum G (k) = (cid:104)q ,q (cid:105) and the total
α α α k
(cid:80)
potential enstrophy spectrum G(k) = G (k). Unlike the case of two-dimensional
α α
Navier-Stokes, where the enstrophy and energy spectra G(k) and E(k) are related via a
simple equation, G(k)=k2E(k), in the generalized layer model, the potential enstrophy
spectrum and the energy spectrum are related indirectly, as shown below:
Define the streamfunction spectrum C (k) = (cid:104)ψ ,ψ (cid:105) . Then, via the properties of
αβ α β k
the bracket above, the energy spectrum E(k) reads
(cid:42) (cid:43)
(cid:88) (cid:88) (cid:88) (cid:88)
E(k)=− (cid:104)ψ ,q (cid:105) =− ψ , L ψ =− L (k)(cid:104)ψ ,ψ (cid:105) (3.10)
α α k α αβ β αβ α β k
α α β k αβ
(cid:88)
=− L (k)C (k), (3.11)
αβ αβ
αβ
and the potential enstrophy spectrum G (k) reads
α
(cid:42) (cid:43)
(cid:88) (cid:88) (cid:88) (cid:88)
G(k)= (cid:104)q ,q (cid:105) = L ψ , L ψ (3.12)
α α k αβ β αγ γ
α α β γ k
8 Eleftherios Gkioulekas
(cid:42) (cid:43)
(cid:88) (cid:88) (cid:88)
= L (k) ψ , L ψ = L (k)L (k)(cid:104)ψ ,ψ (cid:105) (3.13)
αβ β αγ γ αβ αγ β γ k
αβ γ k αβγ
(cid:88)
= L (k)L (k)C (k). (3.14)
αβ αγ βγ
αβγ
Thus,theyarerelatedonlyindirectlyviathestreamfunctionspectrumC (k).Thiswas
αβ
noted previously by Gkioulekas (2012).
We note that for α (cid:54)= β, C (k) is not positive-definite and may take positive or
αβ
negative values. For the case α = β we define U (k) = (cid:104)ψ ,ψ (cid:105) , which is positive
α α α k
definite(i.e.,U (k)≥0).Thenwenotethat2|C (k)|≤U (k)+U (k).Wecanusethis
α αβ α β
inequality to show that if the matrix L (k) satisfies the diagonal dominance condition
αβ
L (k)≥0, for α(cid:54)=β, (3.15)
αβ
(cid:88)
2L (k)+ (L (k)+L (k))≤0, (3.16)
αα αβ βα
αβ
α(cid:54)=β
then the energy spectrum E(k) is positive definite. We give the proof in Appendix A.
Both the two-layer quasi-geostrophic model and the multi-layer quasi-geostrophic model
satisfy this diagonal dominance condition. As for the layer-by-layer potential enstrophy
spectra G (k), it is immediately obvious that they are unconditionally positive definite,
α
regardless of the form of the matrix L (k), since by definition G (k)=(cid:104)q ,q (cid:105) .
αβ α α α k
FortheargumentofthepresentpaperweneedthedissipationratesD (k)andD (k)
E G
for the energy and total potential enstrophy expressed in terms of the streamfunction
spectrum C (k). In appendix B we show that the energy dissipation rate spectrum
αβ
D (k) and the layer-by-layer potential enstrophy dissipation rate spectra D (k) are
E Gα
given by
(cid:88)
D (k)=2 D (k)C (k), (3.17)
E αβ αβ
αβ
(cid:88)
D (k)=−2 L (k)D (k)C (k). (3.18)
Gα αβ αγ βγ
βγ
Note that in order for the dissipation terms to be truly dissipative, the dissipation spec-
tra D (k) and D (k) need to be both positive definite. From the general form of the
E G
above equations this is not readily obvious. However, for simpler configurations of the
dissipation operators, the above expressions for D (k) and D (k) simplify considerably,
E G
thereby making it possible to establish positive definiteness.
Werestrictourattentiontocaseswherethedissipationoperatorsateverylayerinvolve
only the fields of the corresponding layer, with no explicit interlayer terms. This can be
arranged in terms of a linear operator D applied to either the streamfunction ψ or
α α
the potential vorticity q . If D (k) is the spectrum of the positive-definite operator D ,
α α α
then for the case of a dissipation term d = D ψ , we have D (k) = δ D (k). We
α α α αβ αβ β
designate this case as streamfunction-dissipation. The D (k) and D (k) simplify as:
E Gα
(cid:88) (cid:88)
D (k)=2 D (k)C (k)=2 δ D (k)C (k) (3.19)
E αβ αβ αβ β αβ
αβ αβ
(cid:88) (cid:88)
=2 D (k)C (k)=2 D (k)U (k), (3.20)
α αα α α
α α
(cid:88) (cid:88)
D (k)=−2 L (k)D (k)C (k)=−2 L (k)δ D (k)C (k). (3.21)
Gα αβ αγ βγ αβ αγ γ βγ
βγ βγ
the two-layer quasi-geostrophic model 9
(cid:88)
=−2 L (k)D (k)C (k). (3.22)
αβ β αβ
β
Note that for D (k)>0, it follows that D (k)≥0, but it is not obvious that the same
α E
result extends to D (k). However, if we further assume that the same operator is used
Gα
for all layers, i.e. D (k) = D(k), then we have the more specialized case of symmetric
α
streamfunction-dissipation, and the dissipation rate spectra D (k) and D (k) can be
E G
simplified further to give:
(cid:88) (cid:88)
D (k)=2 D (k)U (k)=2D(k) U (k)=2D(k)U(k), (3.23)
E α α α
α α
(cid:88) (cid:88)
D (k)= D (k)=−2 L (k)D (k)C (k) (3.24)
G Gα αβ β αβ
α αβ
(cid:88)
=2D(k)− Lαβ(k)Cαβ(k)=2D(k)E(k). (3.25)
αβ
Now, D(k)>0 implies both D (k)≥0 and D (k)≥0.
E G
Another possible arrangement is potential vorticity dissipation in which the positive-
definiteoperatorD isappliedtothepotentialvorticityq ,therebyyieldingadissipation
α α
term of the form
(cid:88) (cid:88)
d =−D q =− D L ψ = D ψ . (3.26)
α α α α αβ β αβ β
β β
It is easy to see that in this case, D (k) = −D (k)L (k), and consequently, the
αβ α αβ
dissipation rate spectra D (k) and D (k) simplify as follows:
E G
(cid:88) (cid:88)
D (k)=2 D (k)C (k)=−2 D (k)L (k)C (k), (3.27)
E αβ αβ α αβ αβ
αβ αβ
(cid:88) (cid:88)
D (k)=−2 L (k)D (k)C (k)=−2 L (k)[−D (k)L (k)]C (k) (3.28)
Gα αβ αγ βγ αβ α αγ βγ
βγ βγ
(cid:88)
=2Dα(k) Lαβ(k)Lαγ(k)Cβγ(k)=2Dα(k)Gα(k). (3.29)
βγ
Now,forD (k)>0,itfollowsthatD (k)≥0,whereasitisnowthesignofD (k)that
α Gα E
is uncertain. Similarly to the previous case, if we assume symmetric potential vorticity
dissipation,wherethesamedissipationoperatorisusedonalllayers,wethensetD (k)=
α
D(k), and our expression for the dissipation rate spectra D (k) and D (k) simplify to
E G
(cid:88)
D (k)=−2 D (k)L (k)C (k) (3.30)
E α αβ αβ
αβ
(cid:88)
=2D(k)− Lαβ(k)Cαβ(k)=2D(k)E(k), (3.31)
αβ
(cid:88) (cid:88)
D (k)=2 D (k)G (k)=2D(k) G (k)=2D(k)G(k). (3.32)
G α α α
α α
Now, for D(k)>0, it follows again that both D (k)≥0 and D (k)≥0. Note that the
E G
negative sign in −D q was necessary to ensure that the dissipation term d =−D q
α α α α α
remains dissipative when the operator D is positive-definite.
α
10 Eleftherios Gkioulekas
4. Flux Inequality for the two-layer model
We now turn to the main issue of identifying sufficient conditions for satisfying the
flux inequality k2Π (k) − Π (k) ≤ 0 for quasi-geostrophic models. Presently, we re-
E G
strict our interest to the two-layer quasi-geostrophic model under symmetric or asym-
metric streamfunction-dissipation or potential vorticity-dissipation. The multilayer case
and other configurations for the dissipation terms will be considered in future publica-
tions. We begin with a brief review of the two-dimensional Navier-Stokes case and the
formal setup of the argument in section 4.1. Our calculations for the case of streamfunc-
tiondissipationaregiveninsection4.2,andforthecaseofpotentialenstrophydissipation
in section 4.3. Since the details of the calculations are somewhat technical, a summary
discussion of the main results is given in section 5.
4.1. Preliminaries
Let us recall that the energy flux spectrum Π (k) is defined as the amount of energy
E
transferred from the (0,k) interval to the (k,+∞) interval per unit time and per unit
volume.Likewise,thepotentialenstrophyfluxspectrumΠ (k)istheamountofpotential
G
enstrophytransferredfromthe(0,k)intervaltothe(k,+∞)interval,againperunittime
and volume. Assuming a forced-dissipative configuration at steady state and that the
wavenumberk isnotintheforcingrange,theenergyandpotentialenstrophytransferred
into the (k,+∞) interval eventually are dissipated somewhere in that interval. It follows
that we may write the flux spectra Π (k) and Π (k) as integrals of the energy and
E G
potential enstrophy dissipation rate spectra D (k) and D (k):
E G
(cid:90) +∞
Π (k)= D (q)dq, (4.1)
E E
k
(cid:90) +∞
Π (k)= D (q)dq, (4.2)
G G
k
which implies that
(cid:90) +∞ (cid:90) +∞
k2Π (k)−Π (k)= [k2D (q)−D (q)]dq = ∆(k,q)dq. (4.3)
E G E G
k k
We see that a sufficient condition for establishing the flux inequality is to show that
∆(k,q) ≤ 0 for all wavenumbers k < q. It is also easy to see that ∆(k,q) > 0 for all
wavenumbers k <k <q is sufficient for establishing the violation of the flux inequality
t
for all wavenumbers k >k .
t
For the case of two-dimensional Navier-Stokes turbulence, the dissipation rate spectra
D (k) and D (k) are related via D (k) = k2D (k). This immediately gives ∆(k,q) =
E G G E
k2D (q)−D (q) = (k2−q2)D (q) ≤ 0 for all wavenumbers k < q (since D (k) ≥ 0),
E G E E
whichinturngivesthefluxinequalityk2Π (k)−Π (k)≤0.Thephysicalinterpretation
E G
ofthisinequalityisthatwhenwestretchtheseparationofscalesinthedownscalerange,
the energy dissipation rate at small-scales vanishes rapidly. As a result, most of the
injected energy cannot cascade downscale although, as noted by Gkioulekas & Tung
(2005a,b), a small amount of energy is able to do so. As we have seen in the previous
section, for the case of quasi-geostrophic models, the energy and potential enstrophy
dissipation rate spectra no longer have a direct and simple relation with each other, so
the validity of the flux inequality needs to be carefully re-examined.
We set up our analysis of the two-layer quasi-geostrophic model by writing the layer
