Energetics of the layer-thickness form drag based on an integral identity

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Energetics of the layer-thickness form drag based on an
integral identity
H. Aiki, T. Yamagata
To cite this version:
H.Aiki,T.Yamagata. Energeticsofthelayer-thicknessformdragbasedonanintegralidentity. Ocean
Science Discussions, 2006, 3 (3), pp.541-568. ￿hal-00298389￿
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Ocean Sci. Discuss., 3, 541–568, 2006
Ocean Science
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under a Creative Commons License.
Eddy form drag
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open-access review for the journal Ocean Science
H.AikiandT.Yamagata
TitlePage
Abstract Introduction
Energetics of the layer-thickness form
Conclusions References
drag based on an integral identity
Tables Figures
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1 1,2
H. Aiki and T. Yamagata
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1FrontierResearchCenterforGlobalChange,JapanAgencyforMarine-EarthScienceand
Technology,Yokohama-city236-0001,Japan
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2DepartmentofEarthandPlanetaryScience,GraduateSchoolofScience,Universityof
Tokyo,Tokyo113-0033,Japan FullScreen/Esc
Received: 18April2006–Accepted: 30April2006–Published: 20June2006
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Correspondenceto: H.Aiki([email protected])
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Abstract
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The vertical redistribution of the geostrophic momentum by the residual effects of
pressure perturbations (called the layer-thickness form drag) is investigated using 3,541–568,2006
thickness-weighted temporal-averaged mean primitive equations for a continuously
5 stratified fluid in an adiabatic formulation. A four-box energy diagram, in which the Eddy form drag
mean and eddy kinetic energies are defined by the thickness-weighted mean velocity
andthedeviationfromit,respectively,showsthatthelayer-thicknessformdragreduces H.AikiandT.Yamagata
the mean kinetic energy and endows the eddy field with an energy cascade. The en-
ergy equations are derived using an identity (called the “pile-up rule”) between cumu-
lativesumsoftheEulerianmeanquantityandthethickness-weightedmeanquantityin TitlePage
10
each vertical column. The pile-up rule shows that the thickness-weighted mean veloc-
Abstract Introduction
ity satisfies a no-normal-flow boundary condition at the top and bottom of the ocean,
which enables the volume budget of pressure flux divergence in the energy diagram Conclusions References
to be determined. With the pile-up rule, the total kinetic energy based on the Eulerian
Tables Figures
mean can be rewritten in a thickness-weighted form. The four-box energy diagram
15
in the present study should be consistent with energy diagrams of layer models, the
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temporal-residual-mean theory, and Iwasaki’s atmospheric theory. Under certain as-
sumptions, the work of the layer-thickness form drag in the global ocean circulation is J I
suggested to be comparable to the work done by the wind forcing.
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1 Introduction FullScreen/Esc
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In contrast to isotropic three-dimensional turbulence, perturbations in a stratified fluid Printer-friendlyVersion
can induce anisotropic mixing of momentum. The isopycnal (lateral) mixing by the
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Reynolds stress has been well investigated, whereas there have been few investiga-
tions into the diapycnal (vertical) transfer of momentum being possible by the residual
effects of pressure perturbation (called the layer-thickness form drag in this paper, as EGU
25
detailed in Sect. 2.1). Although the layer-thickness form drag has been unpopular in
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modern numerical applications of the ocean and atmosphere, the four-box energy di-
agram shows that the form drag is essential in the connection between the mean and OSD
perturbation fields in an adiabatic formulation of an inviscid hydrostatic fluid.
3,541–568,2006
The four-box energy diagram of ocean and atmosphere dynamics consists of the
potential and kinetic energies associated with the mean and perturbation fields. The
5
classicalLorenz(1955)diagramhasoftenbeenusedinthetheoreticaldevelopmentof Eddy form drag
subgrid-scale parameterization in numerical simulations and in the analysis of various
H.AikiandT.Yamagata
types of data (Bo¨ning and Budich, 1992; Holton, 1992). However, the energy diagram
andassociatedenergycyclemayvarywiththedefinitionsofthemeanandperturbation
fields. An energy diagram for the transformed Eulerian mean (TEM) theory (Andrews
10
TitlePage
andMcIntyre,1976)isgivenbyPlumb(1983)andKanzawa(1984),whereasanenergy
diagram for the generalized Lagrangian mean (GLM) theory (Andrews and McIntyre,
Abstract Introduction
1978) has received little attention in past oceanic studies. Focusing on the adiabatic
aspects of waves and eddies in a stratified fluid, Iwasaki (2001) derived a new energy Conclusions References
diagram from a one-dimensional (vertical direction) analog of the GLM. He showed
15 Tables Figures
that the layer-thickness form drag allows direct transfer between the mean kinetic and
eddy potential energies, which replaces the route involving the eddy kinetic energy in
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theTEMtheory. Moreover,Iwasaki’sformulationdoesnotusethegeostrophicbalance
in closing the energy diagram, which is in sharp contrast to the situation with the TEM J I
theory. This allows Iwasaki’s energy diagram to be applied various types of (rotational
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and nonrotational) stratified fluids. The result of Iwasaki (2001) follows that of Bleck
(1985), who showed that the mean and eddy kinetic energies can be positive-definite
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quantities in isentropic coordinates.
The present study investigated the characteristics of Iwasaki’s energy diagram in
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order to clarify the role of layer-thickness form drag in the connection between the
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mean and perturbation fields, with the aim of understanding the effects of introducing InteractiveDiscussion
layer-thickness form drag in coarse-resolution ocean models (cf. Greatbatch, 1998),
as part of parameterization of unresolved geostrophic eddies in baroclinic instability EGU
(Charney,1947;Eady,1949). Inordertoelucidatethecomponentsrequiredinthenew
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energy diagram, this paper does not use the semi-Lagrangian coordinates of Andrews
and McIntyre (1978), Iwasaki (2001), and Jacobson and Aiki (2006). The present OSD
derivation begins with the inviscid incompressible hydrostatic Boussinesq equations,
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which are adiabatically low-pass filtered so as to avoid unphysical mixing across den-
sitysurfaces. Theseequationsareessentiallythethickness-weighted-meanequations
5
(for tracers, density, and momentum) in density-coordinates (de Szoeke and Bennett, Eddy form drag
1993),asexplainedinSect.2. Wefocusonanintegralidentitytoexplaintheboundary
H.AikiandT.Yamagata
condition (Sect. 2.2). In Sect. 3, we present an energy diagram for the above adiabat-
ically low-pass filtered equations which is largely consistent with the work of Iwasaki
(2001). Under certain assumptions on form-drag parameterization, the work associ-
10
TitlePage
ated with the eddy form drag in the global ocean circulation is estimated in Sect. 4.
The paper concludes with a summary in Sect. 5. The present study excluded diabatic
Abstract Introduction
processes(densitymixing)inthesurfacemixedlayerandthebottomboundarylayerof
theocean(cf.Kuoetal.,2005;PlumbandFerrari,2005),sinceweareconcernedwith Conclusions References
the adiabatic process (mesoscale eddies) and the boundary condition of the present
15 Tables Figures
formulation is clear, in contrast to the TEM theory (see Sects. 2.2 and 3.5).
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2 Adiabatic mean formulation
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Section 2.1 summarizes the thickness-weighted temporal-mean momentum and den-
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sity equations that have been investigated by de Szoeke and Bennett (1993), Mc-
Dougall and McIntosh (2001), and Jacobson and Aiki (2006). Readers not familiar FullScreen/Esc
20
with expressions in z-coordinates are first referred to Bleck (1985) for the primitive
equations(andenergyequations)indensity-coordinates. InSect.2.2,weintroducean Printer-friendlyVersion
integral identity to explain the boundary condition.
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2.1 Primitive equations
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The parameterization of mesoscale eddy transports with an additional advection rep-
resented a major advance in ocean modeling, that allowed coarse-resolution models 3,541–568,2006
to maintain deep water formation in the polar regions and overturning circulation in
5 theworld’soceans(DanabasogluandMcWilliams,1994;Gentetal.,1995;Treguieret Eddy form drag
al., 1997; Killworth, 1997). These theories are based on the thickness-weighted-mean
formulation of a passive tracer equation in density-coordinates (note that “averaging” H.AikiandT.Yamagata
refers to a temporal low-pass filter in this paper). De Szoeke and Bennett (1993)
pointed out in their Appendix A that the mean quantities in density-coordinates can
be mapped back onto z-coordinates (i.e., Cartesian coordinates). That is, a thickness- TitlePage
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ρ ρ
weighted-mean ((∂z/∂ρ)S) /(∂z/∂ρ) is taken in density-coordinates (where S is an Abstract Introduction
arbitrary quantity and ρ is density), which is then mapped back into z-coordinates that
Conclusions References
now refer to the mean vertical position of each isopycnal surface. This backmapped
quantity,nowafunctionofz,isheregiventhesymbolSb(Table1describesthesymbols Tables Figures
usedinthispaper). Apassivetracerequationsuchas∂S/∂t+U·∇S=0inz-coordinates
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becomes, after one application of this process, ∂Sb/∂t+Ub·∇Sb=M[S], where Ub is the J I
thickness-weighted three-dimensional velocity in mean z-coordinates and M[] is the
isopycnal mixing (cf. Griffies, 2004). The weighted three-dimensional velocity is nondi- J I
vergent (∇·Ub=0) if the unweighted three-dimensional velocity is nondivergent (∇·U=0). Back Close
In the special case where the density equation is, M[ρ]=0, diffusion is
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not present. In density-coordinates, the thickness-weighted mean density is FullScreen/Esc
ρ ρ
((∂z/∂ρ)ρ) /(∂z/∂ρ) =ρρ=ρ. As a result, it is useful to introduce Se for an isopycnal
ρ Printer-friendlyVersion
mean (but not thickness-weighted) quantity S that is backmapped onto z-coordinates
at the mean vertical position of each density surface (Table 1). The modified density InteractiveDiscussion
equation (de Szoeke and Bennett, 1993) in z-coordinates becomes
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∂
ρe+Ub ·∇ρe= 0. (1)
∂t
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Herewecallρthemeanheight(MH)density: thisisthesameasthetemporal-residual-
e
mean(TRM)densityinMcDougallandMcIntosh(2001)andJacobsonandAiki(2006), OSD
andisgivenbythedensityofthesurfacewhosemeanverticalpositionisz;itisslightly
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different from the Eulerian mean density ρ (see Fig. 1). In most ocean general circula-
5 tionmodels(OGCMs),thethickness-weightedvelocityUb toadvecttracersiscalculated
by summing the prognostic velocity in the model and a parameterized extra transport Eddy form drag
velocity(detailedinSect.3.3),becauseintheprevailingmeanformulationsthemomen-
H.AikiandT.Yamagata
tum equations are simply averaged either by the isopycnal mean (Gent et al., 1995) or
the Eulerian mean (McDougall and McIntosh, 1996) to avoid modifying the form of the
pressure term.
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TitlePage
However, an interesting feature appears when the momentum equations are also
R
thickness weighted: the hydrostatic pressure gradients −∇ gρ dz(≡G) yield a Abstract Introduction
H z
B
secondary term G (i.e., the layer-thickness form drag, eddy form drag, or invis-
Conclusions References
R
cid pressure drag) in addition to the term available to the model −∇H zgρe dz(≡Ge),
15 where ∇H=(cid:0)∂/∂x,∂/∂y(cid:1). Table 1 provides a detailed expression of GB(≡Gb−Ge). The Tables Figures
thickness-weighted mean momentum equation is
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∂
Vb +Ub ·∇Vb +fz×Vb = Ge/ρ +GB/ρ +M[V], (2)
∂t 0 0 J I
where Vb=(ub,vb) and f is the Coriolis parameter of the earth. The Reynolds stress Back Close
M[V] is less focused on in the present paper, and the total transport velocity Ub has no
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component normal to solid boundaries (McDougall and McIntosh, 2001; see Sect. 2.2
20
for details).
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Equations(1)and(2)firstappearedindeSzoekeandBennett(1993)inanadiabatic
and macroscopic context, and were further investigated in later studies. McDougall
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and McIntosh (2001) introduced a Taylor expansion for the vertical displacement of
densitysurfacesrelativetoz-coordinates. Topresentexactequationsforthemeanand
25 EGU
perturbation fields, Jacobson and Aiki (2006) used a height-density semi-Lagrangian
coordinate that is analogous to the pressure-isentrope semi-Lagrangian coordinate of
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Iwasaki(2001). Equations(1)and(2)arenowwelljustified,beingfreefromexpansion
parameters and the explicit use of density-coordinates, which are improvements over OSD
McDougallandMcIntosh(2001)anddeSzoekeandBennett(1993),respectively. Most
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importantly,velocityUb isaprognosticquantityinamodelsteppingforwardEqs.(1)and
(2),andthissuggeststheapplicabilityofamomentumapproachinwhichthepressure
5
drag GB rather than the eddy-induced advection is parameterized (Greatbatch, 1998; Eddy form drag
Ferreira et al., 2005).
H.AikiandT.Yamagata
2.2 Boundary condition
We consider an oceanic domain bounded by a rigid sea surface and a bottom with TitlePage
10 arbitrary topography. To show that the total transport velocity Ub has no component Abstract Introduction
normal to the boundaries, we here introduce an identity for the vertical integrals of
Eulerian mean and thickness-weighted mean quantities: Conclusions References
Z 0 Z 0 Z ρtop(cid:18) ∂z(cid:19) Tables Figures
S dz = S dz = S dρ =
∂ρ
−h −h ρ
btm J I
ρ
(cid:16) (cid:17)
Z ρetop(cid:18) ∂z(cid:19)ρ Z 0 S∂∂ρz Z 0 J I
S dρe= dz = Sbdz, (3)
ρebtm ∂ρ −h ∂z/∂ρe −h Back Close
where h(>0) is the bottom depth. This identity, which applies to any quantity S, is a
15 FullScreen/Esc
generalization of the results of McDougall and McIntosh (2001) and Killworth (2001),
and is here called the “pile-up rule” since it explains the relations between the cumu-
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lative sums of weighted differentials in the vertical direction. An obvious interpretation
R0
of the pile-up rule is that, with T denoting the range of time averaging, both T S dz InteractiveDiscussion
−h
R0 Rt+T/2R0
and T Sb dz refer to a net amount S dzdt in (z,t) space, measured with
20 −h t−T/2 −h EGU
z-coordinates and density-coordinates, respectively.
Because the no-normal-flow condition of the Eulerian mean velocity U is obvious, it
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isessentialtoshowhowtheremainingpartUb−U (=U+, calledthequasi-Stokesveloc-
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ity in McDougall and McIntosh, 1996, 2001) satisfies the boundary condition (Table 1).
The pile-up rule, Eq. (3), makes the horizontal component of the quasi-Stokes veloc-
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ity purely baroclinic: R0 V+dz (=R0 Vb−Vdz)=0. This allows the overturning stream
−h −h
5 function R−zhV+dz (=R−zhVb−Vdz) to vanish at the top and bottom boundaries, which Eddy form drag
+
confirms the no-normal-flow boundary condition of U . As a result, the total trans-
H.AikiandT.Yamagata
port velocity Ub has no component normal to the top and bottom boundaries, in sharp
contrast to the total transport velocity in the TEM theory.
Explaining the boundary condition becomes less straightforward when the pile-up
TitlePage
rule is not used (Bleck, 1985; Jacobson and Aiki, 2006). The pile-up rule turns out to
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be useful also for the derivation of energy equations (Sect. 3).
Abstract Introduction
Conclusions References
3 Energy equations
Tables Figures
Both the potential and kinetic energies are subject to temporal low-pass filtering, re-
sulting in the so-called total potential and total kinetic energies, respectively, whose J I
equations for inviscid hydrostatic Boussinesq fluids are
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∂
ρgz+∇·(Uρgz) = gwρ, (4)
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∂t
∂ (cid:18)ρ (cid:19) (cid:18) ρ (cid:19) FullScreen/Esc
0|V|2 +∇· U 0|V|2 = V ·G, (5)
∂t 2 2
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where the overbar denotes the Eulerian temporal mean at a constant height. The
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energy interaction is determined by the pressure-flux divergence:
(cid:18) Z (cid:19) EGU
−∇· U ρgdz = V ·G+gwρ, (6)
20 z
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which includes the incompressibility condition ∇·U=0. To simplify the problem, we
consider a volume integral in a closed domain Ω with solid boundaries (i.e., rigid sea OSD
surface). Because the raw velocity U has no component crossing the boundaries of
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the domain, the volume integral of Eqs. (4–6) becomes
d Z Z
ρgz d3x = g wρ d3x, (7) Eddy form drag
5 dt Ω Ω
ρ d Z Z H.AikiandT.Yamagata
0 |V|2 d3x = V ·G d3x, (8)
2 dt Ω Ω
Z Z
0 = V ·G d3x+ gwρ d3x. (9) TitlePage
Ω Ω
Abstract Introduction
In the absence of boundary forcing and friction, the sum of the total potential and total
Conclusions References
kinetic energies is constant.
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3.1 Mean field
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The component of the total energy that is written in terms of resolved quantities, such
as ρe and Vb, is traditionally called the mean energy (a clearer term is the resolved J I
meanenergy). Themeanpotentialandmeankineticenergiesandtheirinteractionare
described by Eqs. (1), (2), and the incompressibility condition ∇·Ub=0: Back Close
∂ (cid:16) (cid:17) FullScreen/Esc
15 ∂t(ρegz)+∇· Ubρegz = gwbρe. (10)
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∂ (cid:18)ρ (cid:19) (cid:18) ρ (cid:19)
∂t 20|Vb|2 +∇· Ub 20|Vb|2 = Vb ·(Ge +GB)+ρ0Vb ·M[V], (11) InteractiveDiscussion
(cid:18) Z (cid:19) EGU
−∇· Ub ρegdz = Vb ·Ge +gwbρe. (12)
z
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