Introduction
The context
In predictability studies, the sensitivity of numerical models to initial
conditions is an important topic. It has been demonstrated in Lorenz's
pioneering work that slightly different initial
states diverge exponentially over time. Thus, theoretical predictability is
often measured by the Lyapunov exponent, which is roughly speaking the
long-term growth rate of the “separation” between neighbouring states
. This characterizes only one aspect, the intrinsic
predictability, of a chaotic system . In practice,
prediction also involves assimilating data to bring the first-guess modelled
state into the “neighbourhood” of the observed state, putting an extrinsic
constraint on predictability. In ensemble prediction methods, a cluster of
close initial model states may be generated around an analysed state to yield
“optimally growing” error structures so as to cover most efficiently the
range of forecast uncertainty and guide targeted observations
.
In the preceding notions of “separation”, “neighbourhood” and “optimally
growing”, the definition of a metric that measures the distance between two
states is fundamental. There are a number of metrics used in the literature
and many authors may hold the view that the definition of a metric is
somewhat arbitrary a priori, especially in the weights given to the
differently dimensioned state variables. The particular choice is often taken
to depend on the application in mind, whether for investigating the
theoretical predictability in a model, estimating optimally growing
perturbations, or minimizing model departures from observations. For example,
if temperature is rather constant in a region, more emphasis may be given to
wind in the metric used to evaluate the theoretical predictability in that
region; if temperature forecast is particularly bad, more emphasis may be put
on temperature in the metric used to generate optimally growing perturbations
or to minimize initial model errors.
In principle, one can adopt any expression to measure the distance between
two points in the phase space of a dynamical system as long as the expression
satisfies the properties of a metric. But only some expressions may have
associated physical significance. For example, geopotential height is often
taken to represent well the wind and temperature in mid-latitude regions
through geostrophic and hydrostatic balance respectively. So in these
regions, the phase space is single-variate and the metric may be defined
simply from the p1 norm, i.e. the domain integral of the absolute
difference in geopotential between two atmospheric states .
In multi-variate phase space, the situation is more complex as there are
various ways of combining state variables into a single metric
. No doubt each of these definitions
have its merits for the purposes they serve. While the value of a metric for
achieving a practical purpose is important in applications, our current work
is mainly concerned with the fundamental theoretical question: is there a
distinguished mathematical formulation of the metric that is consistent with
the intrinsic dynamics of a physical system? Without jumping too much ahead,
the answer lies in having a well-reasoned methodology to formulate such a
metric rather than in a particular form of the metric itself.
Energy-like metrics are most commonly used to measure the distance between
two states of the atmosphere. Some definitions look similar to wave energy
, while others use quadratic expressions
that resemble kinetic and available potential energy
. But none of these
metrics are truly energy or energy differences, as already noted by some
authors (e.g. Ehrendorfer and Errico, 1995). summarized
and compared a number of metrics inspired by expressions of kinetic energy
and total energy. Like energy, enstrophy is another dynamical invariant under
certain conditions and also investigated an enstrophy-like
metric. But the commonality of such approaches lies in (1) the identification
of a dynamical invariant, and (2) the formulation of a metric. The former is
rather well-established in atmospheric dynamical theory; it is the latter
formulation that needs clarification. We shall first review an often used
metric as a concrete illustration of the problem.
An example of a metric
considered dry, compressible flows linearized about a
reference atmosphere at rest with temperature To and surface
pressure po in the absence of surface topography, where To and po are constant in time and uniform in space. The
following integral of quadratic forms over a horizontal domain A is
conserved by the linearized flow when it is adiabatic and inviscid:
ET81=12∫A∫0pou2+v2+cpToT′2dpdA+12∫ARTopop′s2dA,
where cp and R are specific heat capacity at constant pressure
and specific gas constant of dry air. The state variables u, v, T and
ps are zonal wind, meridional wind, temperature and surface
pressure respectively while primes denote perturbations from the reference
state. The tendency of ET81 is a small, time-varying fraction of
the true energy tendency. A derivation of ET81 is given in
Sect. of the Appendix.
As stated in Sect. 4 of , the temperature perturbation
term in ET81 is the available potential energy (APE) of the
system linearized about an isothermal atmosphere. But here we note that it is
not the APE of the real atmosphere which has a non-trivial lapse rate
:
APE=12∫A∫0po1Γd-ΓT-T¯2T¯dpdA,
where Γ is the atmospheric lapse rate, Γd≡g/cp is the adiabatic lapse rate, T¯ is the global
isobaric mean temperature and po=1000 mb.
The associated metric, MT81, proposed by is
given by
MT812=12∫A∫0po(δu)2+(δv)2+cpTo(δT)2dpdA+12∫ARTopo(δps)2dA,
where δ denotes the difference between two evolving atmospheric
states. MT81 is likewise invariant under the linearized dynamics.
The expression defined in Eq. () was originally formulated to study
the convergence of the modelled state to the observed state with repeated
data assimilation cycles in . It was used later by a
number of authors to
measure the evolving difference between two atmospheric states.
Unfortunately, in the latter applications, the uniform To in
Eq. () has lost its physical meaning as a reference state about
which linearization takes place due to large realistic values of lapse rates.
As noted, ET81 and MT81 are not
conserved due to nonlinearity even if realistic flows were adiabatic and
inviscid. We note additionally that significant surface topography like the
Tibetan Plateau, the Rockies and the Andes would also invalidate the
conservation of ET81 in realistic flows and renders questionable
the use of MT81 as a metric. Many authors are probably aware of
these shortcomings but for the lack of a better choice, continue to employ
Eq. ().
At a more fundamental level, while a dynamical invariant is a good metric to
diagnose the change in a system due to data assimilation (which disrupts
model dynamics and hence does not conserve that invariant), it is not a
suitable metric to investigate sensitivity to initial conditions or to search
for optimally growing initial perturbations, precisely because it does not
change during dynamical evolution. The former objective was the subject of
and he succeeded in finding MT81 as such an
invariant metric in linearized flows. The latter two objectives were the
interests of many other authors
who used MT81 or MT81-like metrics, sometimes
appealing to the conditioned conservation of ET81 as a
motivation. But for ET81 to be conserved, the flow has to be
linear which also means that MT81 is invariant and useless for
detecting growing perturbations. This inherent contradiction was not realized
in the literature, no doubt because realistic flows manifest significant
nonlinearity, thus never revealing the otherwise invariant property of
MT81, but also never conserving ET81. This makes
MT81 no more or less justifiable than other metrics, e.g. the
total difference energy of which is MT81 less
the surface pressure contribution. clearly did not
intend his metric to be employed for those latter purposes while the
community continues to use MT81 without a firm theoretical basis.
The essential problem
The essential question that this paper addresses is this: can a metric be
theoretically determined a priori, other than being designed to fit a
particular practical purpose a posteriori? In this theoretical work, we aim
to develop a methodology to construct new non-invariant metrics
based rigorously and consistently on invariant norms. These metrics should
overcome the limitations of having unrealistic reference states and the need
to linearize the flow about those states.
The organization of the paper is as follows. Section
illustrates the methodology by constructing energy-based metrics for the 2-D
barotropic model and for the shallow-water model. Section
follows the same methodology and derives energy-based
metrics for the dry, compressible model in different vertical coordinates. In
Sects. –, the new metrics are
applied to analytic dynamical systems and reanalysis data of the atmosphere.
Finally in Sect. , we discuss the theoretical and
practical advantages of using an invariant norm-induced metric.
Basic methodology for simple fluid systems
Mathematical foundation
A norm on S can be any function ‖•‖:S→[0,+∞)
which satisfies the following properties :
non-negativity, absolute homogeneity, triangle inequality, and is zero only
for the zero vector. Although given the flexibility of defining the norm,
some norms may be interpreted with physical meanings while others may not
when the vector space represents the state of a physical system. For
atmospheric dynamical systems, a natural candidate for the norm is the
square-root of energy which is invariant in the unforced flow. (“Forcing”
here refers generally to diabatic heating, dissipation, mechanical forcing or
gain/loss through domain boundaries.) The importance of the energy-norm is
its conservation property so that any change in the norm means there is a net
forcing or energy flux in or out of the system.
In a normed vector space, the metric between two vectors can be defined as
the norm of the difference between them, and is called the “norm-induced
metric”. But there are other ways of constructing a metric without first
defining a norm, because a metric only needs to satisfy the following
properties: non-negativity, identity of indiscernibles, symmetry and triangle
inequality . When an inner product is defined for a
vector space, the inner product of the difference between two vectors with
itself yields the square of the norm-induced metric. In the literature, both
the norm and the inner
product have been used to define a metric. We have the
following hierarchy :
metricspace⫌normedspace⫌innerproductspace.
In this section, the energy norm and the norm-induced metric are constructed
on the phase space of two simple fluid systems, namely the 2-D barotropic
model and the shallow-water model as an illustration of the basic
methodology.
The two models are assumed to cover a horizontal domain A with periodic
lateral boundary conditions.
2-D barotropic model
For the 2-D barotropic model in Cartesian coordinates (x,y), the kinetic
energy
E=12∫Au2+v2dA,
is conserved, where u=(u,v) is the velocity
vector. The barotropic flow is fully described by the phase vector
x=(u,v), where u and v are functions on the domain A and all
the possible states form the vector space {x∈S}. It is easy to
verify that energy could be used to defined a norm such that
‖x‖2=E. This is the familiar Euclidean norm on the vector space.
Let x1 and x2 be two vectors in S. The norm-induced
metric is defined by
‖x1-x2‖2=12∫Au1-u22+v1-v22dA,
which is similar to the “error kinetic energy” defined by .
For ease of reference, the norm-induced metric can also be called the “separation” in this work.
Shallow-water model
For the shallow-water model, the sum of kinetic and geopotential energy,
E=12∫Ahu2+gh2dA,
is conserved , where h is the height of the water
surface, g is the gravitational acceleration, and the other symbols have
the same meaning as in the barotropic model. Before making use of this energy
expression as a norm, the subtle question is how the phase vector should be
constructed.
The easiest way of constructing the phase space is adding another
“dimension” to the phase space of the barotropic model, which results in a
three-dimensional phase vector (u,v,h).
However, E is not a norm in this vector space because it does not
have the property of absolute homogeneity, i.e. E[μu,μv,μh]≠‖μ‖E[u,v,h], where μ is a real number.
Moreover, since the integrand in Eq. () is not quadratic, despite
being non-negative for a single state, it is not so for the corresponding
difference vector between two states. So we also cannot use Eq. ()
to define a metric for this phase space.
The phase vector is constructed instead as
hu,hv,h so that the energy is the Euclidean norm
in this vector space. Let
x1=h1u1,h1v1,h1 and
x2=h2u2,h2v2,h2 be two
phase vectors. The norm-induced metric or separation, M, is given by
M2=‖x1-x2‖2=12∫A(δhu)2+g(δh)2dA,
where δ denotes taking the difference between the two phase vectors,
e.g. δhu=(h1u1-h2u2).
In mathematics, the axiomatic approach usually defines a norm after the
construction of a vector space. But in physics, we suggest that it is more
useful to first identify the physical quantity which we desire to be the norm
and then try to construct the vector space such that this quantity is indeed
a norm on that vector space. In the above example, we have adopted the latter
approach and constructed the vector space such that energy is the familiar
Euclidean norm again. The norm-induced metric follows naturally thereafter.
Linearization of separation
Let the norm be E=fa,b+ga,b, where f and g
are positive definite functions of variables a and b, so that E is the
Euclidean norm on the vector space x∈S|x=f,g. If variables a and b
are observed and recorded in practice rather than f and g, it is more
convenient to transform f,g coordinates to the
more conventional a,b coordinates. Although the relation
between the two coordinate systems may involve nonlinear transformations, the
increments δf,δg can be approximated
by linear combinations of δa,δb assuming the
increments are small . One example is the transformation
from Cartesian coordinates x,y,z to spherical polar
coordinates λ,ϕ,r where λ is longitude,
ϕ is latitude and r is the distance from origin .
The norm-induced metric in f,g coordinates
is given by
M2=δf,δgδf,δgT.
Using the total increment theorem, M can be approximated in the tangent
space y∈T|y=δa,δb
by
M2≈δa,δbGδa,δbT,
where the metric tensor
G=14fa2/f+ga2/gfafb/f+gagb/gfafb/f+gagb/gfb2/f+gb2/g,
where the subscripts denote derivatives with respect to that variable. Notice
that when f=a2 and g=b2, the metric tensor is simply the identity
matrix and Eq. () reproduces the Euclidean norm.
In the shallow-water model, we may transform
hu,hv,h coordinates to the more conventional
(u,v,h) coordinates. If two vectors x1 and x2 are
close to each other, the total increment δhu can be linearized
about a reference state which could be either one of the two vectors. Hence,
Eq. () can be rewritten as
M2≈12∫Aδu,δv,δhh0u/20hv/2u/2v/2u2+v2/4h+gδuδvδhdA,
which is non-Euclidean as the metric tensor matrix is not diagonal. The
separation-squared between two neighbouring states is linearized about one of
them but importantly, the dynamics governing the evolution of both states
remain nonlinear.
Dry compressible atmosphere
In this section, the definition of the separation metric is extended using
the same methodology as above to fully compressible equations of a dry,
adiabatic and inviscid atmosphere in different vertical coordinates. The
horizontal domain A is assumed to be closed or periodic.
Pressure coordinate
The formulation of a metric in pressure coordinate p is useful because much
observation and reanalysis data are presented on pressure levels. In
p-coordinate, it can be proven that the following quantity is conserved
:
E=1g∫A∫0pH12u2+cpT+ΦHdpdA,
where T is temperature, ΦH is surface geopotential, pH is surface pressure, cp is the specific heat capacity of dry
air at constant pressure, and the other symbols are as
before. The energy in Eq. () is not a norm in the vector space
(u,v,T) as pH also appears as a variable in the upper
limit of the first integral. This means that Eq. () cannot be used
directly to define the norm-induced metric.
To make further progress, consider a constant reference pressure pr close to but smaller than pH such that the vertical
integration of the first two terms in Eq. () could be separated
into a main contribution 0,pr and a boundary-layer
contribution pr,pH. In the boundary layer,
the kinetic energy u2/2 is always much less than cpT
and the temperature does not deviate much from a reference temperature
Tr(x,y), which could be conveniently defined by the vertical
gradient of a hydrostatically balanced geopotential field
Φref(x,y,p) at pressure pr:
Tr=-prR∂Φref∂pp=pr.
So Eq. () can be approximated by
E≈1g∫A∫0pr12u2+cpTdpdA+1g∫AcpTrpH-pr+ΦHpHdA,
where (pH-pr) represents the boundary-layer mass. Note
that the atmosphere stays close to the reference state
Φref(x,y,p) because hydrostatic balance must be dominant for
the pressure coordinate to be reasonably employed. This implies that surface
pressure and boundary-layer temperature must always stay close to their
reference values. So the above approximation is as good as the hydrostatic
balance implicitly assumed in pressure coordinate and a modified energy
expression differing by a constant from Eq. () can be defined:
Emod=1g∫A∫0pr12u2+cpTdpdA+1g∫AcpTr+ΦHpHdA.
The energy expression Eq. () is used to define the norm with the
phase vector defined as x=u,v,T,pH since ΦH and Tr are time independent.
The separation metric of compressible flows in pressure coordinate is given
by
M2=1g∫A∫0pr12δu2+cpδT2dpdA+1g∫AcpTr+ΦHδpH2dA,
where the three contributions by differences in wind, temperature and surface
pressure are henceforth called kinetic, enthalpy and surface pressure
components of separation-squared respectively.
Equation () could be approximated in terms of perturbations of the
more conventional variables u, v, T and pH as
M2≈1g∫A∫0pr12δu2+cpδT24TdpdA+1g∫AcpTr+ΦHδpH24pHdA.
The approximation in Eq. () only requires δT/T≪1 because
hydrostatic balance then implies δpH/pH≪1.
The separation metric in Eq. () is linearized in the sense that it
has been transformed into the tangent linear space at (T,pH). It
is different from MT81 in Eq. () in the coefficients of
temperature difference δT2 and surface pressure
difference δpH2 by factors 2 and
2R/cp respectively. In our expression, the reference state is
realistic and can evolve nonlinearly with time. We also account for the
influence of surface topography. Moreover, no linearization is assumed in the
flow dynamics in developing Eq. () and a more accurate expression
for the separation metric is available in Eq. () for atmospheric
states that are not close, i.e. δT/T≳0.1, or less likely,
δpH/pH≳0.1.
Isentropic coordinate
The use of potential temperature as a vertical coordinate dates back half a
century when for example, Lagrangian parcel trajectories were traced on
isentropic surfaces . further proposed a
potential vorticity – potential temperature view of the general circulation
which has advantages in understanding atmospheric dynamics and advanced
mid-latitude weather forecasts. Thus, it is of both theoretical and practical
interest to examine our separation metric in isentropic coordinate.
In isentropic coordinate θ, the conserved energy in Eq. ()
takes the form
E=∫A∫θH∞12u2+θΠ+ΦHσdθdA,
where σ=-g-1∂p/∂θ is isentropic density,
Π=cpp/1000mbR/cp is Exner's function,
θH is surface potential temperature, and the other symbols are as
before. Similar to the case of pressure coordinate, we define an appropriate
reference potential temperature at the lower boundary
θrx,y to separate the main contribution and the
boundary-layer contribution as
E≈∫A∫θr∞12σu2+θσΠ+σΦHdθdA+∫A∫θHθrθΠ+ΦHσdθdA,
where we have ignored the kinetic energy in the boundary layer as before.
Since θΠ=cpT and σdθ=ρdz, where ρ is
mass density and z is height, the boundary-layer term can be evaluated as
∫θHθrθΠ+ΦHσdθ=∫zHzθr1+ΦHθΠcpRpdz≈1+ΦHθrΠrcpRprzθr-zH,
where zH is the surface topography, and zθr=z(x,y,θr,t) is the elevation of the
θr-surface as further elaborated in
Sect. . The reference boundary-layer pressure pr(x,y) and Exner's function Πr(x,y) are defined from the
vertical gradient of the hydrostatically balanced Montgomery potential field
Mref(x,y,θ) at isentropic level θr:
Πr≡cppr/1000mbR/cp=∂Mref∂θθ=θr.
This allows a modified energy expression differing from Eq. () by
a constant to be defined:
Emod=∫A∫θr∞12σu2+θσΠ+σΦHdθdA+cpR∫A1+ΦHθrΠrprzθrdA.
We define the phase vector as
x=σu,σv,σΠ,σ,zθr
so that the separation metric in isentropic coordinate induced by the
Euclidean norm in Eq. () is given by
M2=∫A∫θr∞12δσu2+θδσΠ2+ΦHδσ2dθdA+cpR∫A1+ΦHθrΠrprδzθr2dA.
Compared to the separation metric in pressure coordinate, Eq. ()
depends on one more thermodynamic variable, the isentropic density σ,
because the flow is compressible in isentropic coordinate whereas it is
non-divergent in pressure coordinate. The linearized separation-squared in the
tangent linear space (u,v,Π,σ,zθr) is given by the
non-Euclidean form
M2≈∫A∫θr∞σ2δu2+θσδΠ24Π+u2/2+θΠ+ΦHδσ24σ+u⋅δu+θδΠδσ2dθdA+cpR∫A1+ΦHθrΠrprδzθr24zθrdA.
The variation of the elevation of the isentropic surface θr
can be further related to the variation of the surface potential temperature
at each location (x,y):
δzθr≈-δθHΘzr,Θzr=-gθr∂2Mref∂θ2θ=θr-1,
where Θzr(x,y) is the reference (positive) static stability in the boundary layer defined consistently above as
∂θ∂z∂2M∂θ2=∂θ∂z∂Π∂θ=∂p∂zdΠdp=-gρRΠcpp=-gΠcpT=-gθ.
Geopotential height coordinate
Both pressure and isentropic coordinate formulations above are limited by
their underlying assumption of hydrostatic balance. Current numerical weather
prediction (NWP) models are able to model non-hydrostatic flows at mesoscale
resolution, and many predictability studies are conducted based on NWP model
results . Therefore, it is useful to
derive the separation metric without making the hydrostatic assumption and
here we make use of the geopotential height coordinate.
In geopotential height coordinate, z≡Φ/g and total energy is the
sum of kinetic energy, internal energy and geopotential energy
E=∫A∫zH∞12ρv2+ρcvT+ρgzdzdA,
where v=u,v,w is the 3-D velocity, cv is the
specific heat capacity for dry air at constant volume, and the other symbols
are as before. For an ideal gas, Eq. () can be rewritten as
E=∫A∫zH∞12ρv2+cvRp+ρgzdzdA.
By defining the phase vector as
x=ρu,ρv,ρw,p,ρ,
Eq. () specifies a Euclidean norm. Hence, the separation metric is
given by
M2=∫A∫zH∞12δρv2+cvRδp2+gzδρ2dzdA.
Actually, M is not dependent on the precise set of variables E is
expressed in. The separation metric induced by Eq. () is the
equivalent to that induced by Eq. (). Compared to the separation
metric in pressure and isentropic coordinate, Eq. () depends on
w because the flow is non-hydrostatic. Note that the absence of a
boundary-layer term in Eq. () is because the bottom boundary is
rigid in geopotential height coordinate.
Equation () can be linearized and approximated in
(u,v,w,p,ρ)-space as
M2≈∫A∫zH∞ρ2δv2+cvRδp24p+v2/2+gzδρ24ρ+v2⋅δvδρdzdA.
The linear approximation requires the fractional difference δp/p and
δρ/ρ to be much smaller than one. The terms above are physical
analogues to those in Eq. () for the shallow-water model, apart
from the additional internal energy term. Equation () can be
further simplified to
M2≈∫A∫zH∞12ρδv2+cvRδp24p+gzδρ24ρdzdA,
if |δv|/|v|≫|δρ|/ρ holds true over most of
the integration domain.
Generalized coordinate and finite upper boundary
In the preceding sections, the upper boundary of the atmosphere is always
assumed to be at zero or infinity. But it is impossible to span the whole
atmosphere in a numerical model and a finite upper boundary is prescribed. In
this section, we treat the case of a generalized vertical coordinate with
finite upper and lower boundaries. The two boundaries are assumed to be
material surfaces to conserve the mass between them.
It has been shown that energy for a dry,
compressible atmosphere in generalized vertical coordinate s takes the
form
E=∫A∫sHsTu2+v2+ϵvw2/2+cvT+ΦσsdsdA,
where σs=ρ∂z/∂s, Φ is the
geopotential and δv is the switch between non-hydrostatic
(ϵv=1) and hydrostatic (ϵv=0) flows. The
subscripts H and T denote values at the lower and upper boundaries
respectively and the other symbols are as before. The integrand is similar to
that in geopotential height coordinate except that the density is multiplied by
the Jacobian of the vertical coordinate transformation.
E is conserved only if the upper boundary is a rigid lid, i.e. zT=zT(x,y), so that no work is done there. But this boundary
condition is not realistic for the atmosphere. Instead, we consider the case
of an “elastic lid”, i.e. pT= constant, where an energy-like
invariant exists . For a non-hydrostatic atmosphere,
this invariant is
E=∫A∫sHsTσsu2+v2+w2/2+σscvT+σsΦ+pTJ)dsdA,
where J=∂z/∂s is the Jacobian of the vertical coordinate
transformation. The last term in the integrand arises from work done at the
upper boundary. For a hydrostatic atmosphere, the energy-like invariant in
Eq. () is simplified by combining pressure work, internal energy
and gain in geopotential above the surface into enthalpy and dropping away
the vertical velocity contribution to get
E=∫A∫sHsTu2+v2/2+cpT+ΦHσsdsdA.
Note that the energy density in Eq. () reproduces the energy
density in pressure and isentropic coordinates with zero and infinite upper
boundary respectively (see Eqs. and ).
As before, Eqs. () and () can be approximated by
decomposing E into an integral with constant integration limits [sL,sU] over the main atmospheric body and boundary-layer
contributions over [sH,sL] and [sU,sT]. To make use of the rigid lower boundary condition, we integrate with
respect to z over the lower boundary layer (except for s≡p, see
Sect. ). Likewise, to make use of the elastic upper
boundary condition, we integrate with respect to p over the upper boundary
layer. We make the hydrostatic approximation in both boundary-layer
integrations because hydrostatic balance is still dominant in the atmosphere
even when the flow is non-hydrostatic. Thus, modified energy expressions for
the non-hydrostatic and hydrostatic atmosphere, Emodnh
and Emodh respectively, can be defined after dropping
away constant contributions:
Emodnh=∫A∫sLsUσsu2+v2+w2/2+cvσsT+σsΦ+pTJdsdA+∫AcvTL+ΦH+pTρLρLzLdA+1g∫AcpRpTρU+ΦUpUdA,Emodh=∫A∫sLsUσsu2+v2/2+cpσsT+σsΦHdsdA+∫AcpTL+ΦHρLzLdA+1g∫AcpRpTρU+ΦHpUdA,
where ρL and TL are reference functions at sL, while ρU and ΦU are reference functions at
sU, all of which are functions of (x,y) only. The zL is
the elevation at sL and pU is the pressure at sU. So the phase vectors xnh and xh
respectively for the non-hydrostatic and hydrostatic atmosphere are defined as
xnh=|σs|u,|σs|v,|σs|w,|σs|T,|σs|Φ,|J|,zL,pU,xh=|σs|u,|σs|v,|σs|T,|σs|,zL,pU,
and the norm-induced metrics can be defined as before. For a non-hydrostatic
atmosphere, the degree of freedom |J| in Eq. () compensates for
the loss in internal energy due to work done by the atmosphere at the upper
boundary. But |J| is not a degree of freedom for a hydrostatic atmosphere.
The reason is that when pressure p is kept fixed, cp(dT)p≡(ðQ)p so that enthalpy in
Eq. () can only be changed by heat transfer and is invariant to
work done at the upper boundary.
In geopotential height coordinate (s≡z), the Jacobian J is
identical to one and so drops out from the phase vector of a non-hydrostatic
atmosphere, while Φ is a function of the coordinate only and so
ρ and not ρΦ is the phase coordinate. The
lower-boundary coordinate is time-independent (zL≡zH) and so the lower-boundary integrals are constant and can be dropped from
Eqs. () and (). So our generalization is consistent
with the results of Sect. .
In pressure coordinate (s≡p), the “density” |σs|=1/g is a constant and so drops out from the phase vector. The
upper-boundary coordinate is constant (pU≡pT) and
so the upper-boundary integrals are also constant and can be dropped from
Eq. (). So the separation metrics in Eqs. () and
() in pressure coordinate are still valid for an atmosphere with
an elastic lid, although the vertical integrals start from pT
instead of zero.
When the atmosphere has an elastic lid, the metrics for geopotential height
and isentropic coordinates have additional upper-boundary terms while the
vertical integrals over the main atmospheric body have finite constant upper
limits, unlike Eqs. () and (). The pressure change
δpU on the constant upper-limit coordinate surface is
directly related to the movement of the elastic lid as follows:
δpU(zU)≈gρUδzT,
where ρU is the reference mass density at zU in
geopotential height coordinate:
δpU(θU)≈gρUΘzUδ=gcpRpTΠTδθTθUΘzU
where ΘzU is the reference static stability at θU
defined similarly to Eq. () and ΠT is the
constant Exner function on the elastic lid in isentropic coordinate.
Elevation at the top of the lower boundary layer
Geopotential is assigned to be zero at the lowest point on Earth's surface
for energy to satisfy the non-negative requirement of a norm. So the null
vector corresponds to a state where Earth's surface is flat. This zero-point
is not arbitrary as it is not possible to extract any more geopotential
energy from this point by moving air around. So by definition,
geopotential height z is also zero at the lowest point on Earth's
surface.
The zL in Eqs. () and () (or zθr in
Eq. () for θ-coordinate) is the elevation of the
coordinate surface sL at the top of the lower boundary layer (or
θr for θ-coordinate). Elevation is really just a
geometric coordinate that can have a zero-point at any level. Thus an
arbitrary constant can be added to Emodnh or
Emodh so that zL is arbitrary to a constant
value. In other words, zL need not be identical to the
geopotential height z at sL as is implicitly assumed so far.
This means that our theoretical formulation of the norm and norm-induced metric is
not yet complete at this juncture. (There is no corresponding problem at the
upper boundary because pU is a thermodynamic coordinate of which
the zero-point is well-defined physically.)
In pressure coordinate, we treat the lower boundary differently, integrating
with respect to p instead of z. The basic reason is that the formulation
is simpler in pressure coordinate: the phase space has one less state
variable (as “density” is a constant in pressure coordinate), the top of
the boundary layer, pr, is constant and the phase coordinate
arising from the lower-boundary contribution is a well-defined thermodynamic
function, pH. Moreover, our formulation in Eq. () has
the advantage of sharing essentially the same form as the already widely used
MT81 in Eq. () (apart from factors of 2 and
2R/cp in the temperature and surface pressure terms and the
inclusion of surface topography). In contrast, integrating with respect to
z leads to a phase coordinate zL where zL is
an arbitrary constant and in general, one must assume reference values for
two thermodynamic functions (TL and ρL) instead of
one.
The uniqueness of the pressure coordinate allows us to calibrate zL by implementing Eq. () for s≡p and requiring that the
lower boundary-layer contribution to the energy norm be equal to that in
Eq. (). Hence,
pHg=ρLzL+zo,
where zL is the geopotential height at sL as before and
zo is the arbitrary constant to be calibrated. From hydrostatic
balance, to first order,
pH-pr=gρLzL-zH,
where pr is the time-independent reference pressure at
the top of the boundary layer in pressure coordinate as defined in
Sect. . Equations () and ()
imply
zo=-zH+prgρL=-zH+RTrg,
where Tr is the reference boundary-layer temperature in pressure
coordinate as defined in Sect. .
Now we define a generalized “local elevation”
ZL=defzL-zH+RTLg,
which is the elevation from a zero-point located locally at a distance
RTL/g below the surface. RTL/g is the density
scale height derived from the reference lower boundary-layer temperature
TL(x,y) in the generalized coordinate. The zero-point of local
elevation is shallower underground in regions of high terrain. When we add
the relevant constants to Emodnh and
Emodh, the phase coordinate arising from the lower
boundary condition becomes the locally calibrated ZL
instead of the globally calibrated, geopotential-based zL.
Note that the definitions of geopotential and geopotential height are not
affected by this local calibration.
Application of the calibration in isentropic coordinate leads to zθr being replaced by
Zθr=defzθr-zH+RgcpθrΠr,
which is the local elevation of the θr-surface in
Sect. . This fixes the hidden problem in
Eq. (), and hence in Eq. (), where the boundary-layer
contribution could be arbitrarily small because zθr in the
denominator of the integrand is arbitrary to a constant. In pressure
coordinate, the local elevation actually measures surface pressure as it can
be shown that ZL=(pH/pr)(RTr/g). Our
theoretical formulation is now complete and consistent among all coordinates.
Example I: geostrophic balanced flow of shallow-water model
In this section, the separation metric of the shallow-water model is applied
to an axisymmetric geostrophically balanced flow in polar coordinates in a rotating frame.
Let r and λ be the radial and angular coordinates respectively,
and u and v be the radial and azimuthal
velocity components respectively, and the rest of the symbols follow the same notation
as in previous sections. The flow is initially at rest with height given
by
ho=-ho′+Hor<a1Hor>a1,
where Ho is the basic height and ho′ is the initial
disturbance. The initial potential vorticity profile is
ξ=f+ζh=f-ho′+Hor<a1fHor>a1,
where ξ is potential vorticity (PV), f is the Coriolis parameter (or
“planetary” vorticity), and ζ=r-1∂rv/∂r is
the relative vorticity. The geostrophic balanced state can be solved
analytically by PV conservation without assumption of ho′≪Ho as pointed by , where he gave the non-dimensional
solution of geostrophic adjustment in Cartesian coordinates. The boundary
conditions are: ∂h/∂r=0 at r=0 and h=0 as
r→∞, which means the azimuthal velocity at the origin is
zero and the perturbation dies away at infinity.
For simplicity, the following non-dimensional variables are introduced:
h′̃=hHo-1,r̃=rLd,ṽ=vfLd,ξ̃=ξf/Ho,
where h′̃ is the non-dimensional perturbation height and
Ld=gHo/f is the Rossby radius of deformation. As finally
axisymmetry and geostrophic balance is restored, ũ=0 and
ṽ=∂h′̃/∂r̃. Since the
initial disturbance could be strong, it is necessary to consider the
advection of fluid columns from initial positions . Let
a2 be the new PV discontinuity point, i.e. a2-a1 is the
displacement of the PV boundary. Then the non-dimensional conservation of PV
becomes
∂2h′̃∂r̃2+1r̃∂h′̃∂r̃+1h′̃+1=1/ηr̃<a2/Ld1r̃>a2/Ld,
where η=-ho′/Ho+1. The solution is determined by
parameters ho′/Ho and a2/Ld.
Solving Eq. () separately for r̃<a2/Ld and
r̃>a2/Ld and matching the solutions at
r̃=a2/Ld gives the balanced perturbation height
h′̃=-ho′Ho1-I0r̃/ηMa2/Ldr̃≤a2/Ld-ho′HoK0r̃Na2/Ldr̃>a2/Ld,
where Iα and Kα are modified Bessel functions
of the first and second kind, and
M(x)=I0x/η+1ηK0xK1xI1x/η,N(x)=K0x+ηI0x/ηI1x/ηK1x.
The balanced velocity can be obtained as
ṽ=ho′Ho1ηI1r̃/ηMa2/Ldr̃≤a2/Ldho′HoK1r̃Na2/Ldr̃>a2/Ld,
where a2 is determined by the mass conservation equation
∫0∞h′̃r̃dr̃=∫0a1/Ld-ho′/Hor̃dr̃.
The left-hand side of Eq. () is a monotonic decreasing function of
a2, hence the solution of a2 is unique and increases as a1
increases.
The non-dimensional separation metric in polar coordinates is given by
M2=12∫0∞δh̃ṽ2+δh̃2r̃dr̃,
which linearizes as
M2≈12∫0∞ṽδṽδh̃+h̃δṽ2+ṽ24h̃+1δh̃2r̃dr̃,
where the first three quadratic terms involving δṽ and ṽ sum up to approximate the kinetic separation-squared.
The non-dimensional PV profile is shown in Fig. a with
a1/Ld=4 and ho′/Ho=0.8. Figures b and
c show the non-dimensional solutions of height and tangential
velocity. The PV boundary is displaced from r̃=4 to
r̃=3.53. The low-PV water mass originally at region D now moves
to B and C and pushes the high-PV water mass originally at C to A. The
tangential velocity maximizes at the new PV boundary a2. The Rossby
number Ro=V/fL=ṽLd/a2 is about 0.17 which is
small so that the geostrophic approximation is good.
Non-dimensional solution of PV, height and tangential velocity with
a1/Ld=4 and ho′/Ho=0.8 in the geostrophically
balanced shallow-water model.
The ratio between the mixed term and the other two quadratic terms
of the linearized kinetic separation-squared when a1/Ld=4 and
(a) ho′/Ho=0.1 and (b) ho′/Ho=0.8. The value at the origin is not defined.
In order to investigate the importance of the mixed term in
separation-squared, two sets of balanced solutions with different initial
height discontinuity but the same initial radius of high PV (a1/Ld=4)
are investigated. The first case is ho′/Ho=0.1, where the
flow is more like a linear system. The second case is ho′/Ho=0.8, where the flow is nonlinear. Separation metrics are calculated by
adding perturbations to the two control parameters a1/Ld and
ho′/Ho for both cases. All perturbations are sufficiently
small that linearization of separation-squared is good. The ratio between the
mixed term
12∫0∞ṽδṽδh̃r̃dr̃
and the other two quadratic terms of the linearized kinetic
separation-squared for both cases are shown in Fig. .
(a) The non-dimensional kinetic separation-squared when
a1/Ld=4 and ho′/Ho=0.8. (b) Fractional error of
linearized kinetic separation-squared from (a). The white dot at the centre
marks the origin.
Balanced (a) temperature (K) and (b) zonal wind (m s-1)
in the 2-D thermal wind model. (c) Ratio of the non-dimensional kinetic
to enthalpy components of separation-squared when Δh and
Δp̃ are perturbed.
In the first case of almost linear flow, the mixed term's contribution is
always less than 5 % of the δṽ2 term's
contribution. Hence, the non-Euclidean separation metric can be approximated
by ignoring the mixed term. But in doing so, one must also ignore the kinetic
enhancement of the δh̃2 term as it is generally even
smaller. This is because the flow has only small PV differences which induces
small velocities and so all terms involving ṽ must be
consistently ignored.
In the second case of nonlinear flow, the contribution of the mixed term
could be comparable to that of the δṽ2 term unless the
perturbations are almost entirely in the extent a1/Ld rather than the
magnitude ho′/Ho of the initial low-PV fluid. Here, it is
also generally inconsistent to keep the kinetic enhancement of the
δh̃2 term without keeping the mixed term. Therefore, in
nonlinear flows, the non-Euclidean characteristic of the linearized metric
cannot be neglected because large PV differences lead to large velocities
ṽ.
The kinetic separation-squared (the first term of Eq. ) for the
nonlinear flow where ho′/Ho=0.8 and the fractional error
made in using the linearized expression (the first three terms in
Eq. ) are shown in Fig. . Note that the size of
perturbations on a1/Ld and ho′/Ho is comparable
to the parameters themselves. The linearized separation-squared is only valid
for very small perturbations, or for a special subset of the dynamical
parameters.
Example II: 2-D thermal wind model in pressure coordinate
In this section, the separation metric of a dry compressible atmosphere in
pressure coordinate is applied to a 2-D thermal wind flow in the Northern
Hemisphere. The zonal wind u at the surface is assumed be to zero. The
potential temperature θ under radiative equilibrium is assumed to be
θθo=1-Δhsin83ϕ2expp-pch/Δp-12expp-pch/Δp+1+Δvlnpop,
where ϕ is latitude, po is the constant surface pressure,
θo is the constant surface temperature at the equator,
Δh and Δv control the fractional change of
potential temperature from equator to pole and from the surface to the
tropopause respectively, pch controls the pressure where the
equator–pole temperature difference changes sign, Δp is a
factor controlling the vertical extent of the balanced jet and the other
symbols follow the same notation as in previous sections. The form of
Eq. () is inspired by the work of .
It is convenient to introduce the following non-dimensional variables:
T̃=Tθo,ũ=uRθo/Ωea,p̃=ppo,Ẽ=Ecpθopoa/g,
where Ωe is the angular speed of rotation of Earth, a is the
radius of Earth, and the rest of the symbols follow the same notation as
Sect. . So the non-dimensional thermal wind equation can
be written as
2sinϕ∂ũ∂p̃=1p̃∂T̃∂ϕ,
Given the equilibrium potential temperature, the non-dimensional temperature
is
T̃=p̃R/cp1-Δhsin83ϕ2expp̃-p̃ch/Δp̃-12expp̃-p̃ch/Δp̃+1-Δvlnp̃,
where Δp̃ is the non-dimensionalized Δp.
The solution for zonal wind is obtained by integrating the right-hand side of
Eq. () from the surface to p̃. The non-dimensional
separation metric is given by
M2=∫-π2π2∫01R2θocpΩe2a2δũ22+δT̃2cosϕdp̃dϕ,
where we have used the fact that the depth of the atmosphere is much
smaller than the radius of Earth. Notice that T̃ and
ũ are both well-defined finite functions of p̃
over 0, 1 so the integral is finite.
The following parameters are specified: θo=300 K, po=1000 mb, pch=220 mb, Δv=35/300. We consider
a reference solution with Δh=40/300 and
Δp̃=50/1000. Figure a and b show the balanced
temperature and zonal wind profiles of the reference solution. The westerly
jet is formed around 40∘ N with a maximum velocity of 21.5
m s-1 at about 200 mb. The thermal wind balance model is
more valid in the mid-latitudes and hence the solution in the tropics is not a
good approximation of the real atmosphere. Another deficiency is the vertical
temperature profile in this model does not describe the temperature inversion
above tropopause. Therefore we shall only make use of the data below 100 mb
and between 35–65∘ N.
Separation metrics are calculated when Δh and
Δp̃ are perturbed. From the results in
Fig. c, the kinetic component is always less than the enthalpy
component though the ratio between them varies with perturbations. The order
of magnitude of the ratio of kinetic to enthalpy components is set
fundamentally by the ratio of temperature and specific angular momentum
parameters on Earth, θo/(Ωea)2. In the next
section, the relative importance of kinetic and enthalpy components of
separation-squared is further investigated with reanalysis data.
Example III: reanalysis data of the atmosphere
The separation metric of a dry compressible atmosphere in pressure coordinate
is also applied to the reanalysis data. The data used in this study are the NCEP
Climate Forecast System Reanalysis (CFSR) monthly mean of 6-hourly forecasts
. The reanalysis monthly mean data
cover 31 years from January 1979 to December 2009 with
0.5∘×0.5∘ spatial resolution and 37 vertical levels from
1000 mb to 1 mb. Temperature, zonal and meridional wind at 37 pressure
levels as well as geopotential height, pressure and temperature at the
surface are used for the calculation. The separation metric Eq. ()
is transformed from Cartesian coordinates to spherical coordinates with the
Jacobian r2cosϕ. Since the depth of the atmosphere is much smaller
than the radius of Earth, the Jacobian can be approximated by
a2cosϕ.
For the work here, at each grid point, we chose pr to be the
smallest surface pressure ever attained and linearly interpolate for
Tr at pr between the mean temperature at the lowest
pressure-level above the surface and the mean surface temperature (assumed to
be at the mean value of surface pressure) in the data set. A quick check with
the data set shows that temperature within 100 mb from the surface never
deviates by more than 2.5 % and so the approximation T≈Tr in the boundary layer in Eq. () is valid. We also
confirmed that linearization of the separation-squared in Eq. ()
is justified.
(a) Kinetic (red), enthalpy (green) and surface pressure
(blue) separation-squared between CFSR zonal mean monthly mean data and its
annual mean climatology in mid-latitudes (35–65∘ N) up to
100 mb. (b) As (a), but using full 3-D data
including the eddies. (c) Enthalpy separation-squared calculated
with different formulae: as the author proposed (green solid line), using
constant temperature To=270 K in the integrand cpδT2/4To (black dashed line),
further multiplying by a factor of 2 to get cpδT2/2To (cyan dashed line), and using constant
po=1000 mb instead of pr(x,y) for the integration
upper limit but excluding isobars below the surface (grey solid line). Note
that the vertical scale is logarithmic in (a, b) but is linear in
(c).
We investigate the separation between the monthly mean state of the
atmosphere represented by CFSR data and its annual mean climatology. The
annual mean climatology is defined as the mean over all months in 31 years of
CFSR data and so is time independent. It provides the values for T and
pH in the denominators of the terms in Eq. ().
Mid-latitude zonal mean and eddies
The separation-squared between zonal mean CFSR monthly mean data and its
annual mean climatology in mid-latitudes (35–65∘ N) up to 100 mb
between 2001 and 2009 is shown in Fig. a. The averaging interval
for zonal mean is confined to isobars above the surface. Kinetic and enthalpy
components show a synchronous semi-annual oscillation, which maximizes in
January or February and in July or August. This is consistent with the
seasonal cycle: the atmosphere moves furthest from the annual mean state
during winter and summer. The surface pressure component is noisier and the
semi-annual oscillation is not obvious. The reason is that surface pressure
has strong zonal asymmetry due to the distribution of continents and oceans
and the seasonal cycles of surface pressure are out of phase between
continents and oceans.
(a) Surface elevation zH in the northern mid-latitudes
(35–65∘ N). Surface pressure separation-squared in Dec
2005 contributed by (b) surface topography, zHδpH2/4pH, and (c) enthalpy,
1gcpTrδpH2/4pH. (d) The enthalpy contribution using
Talagrand's formula, 1gRToδpH2/2po, where To=270 K and
po=1000 mb, for comparison with (c).
The kinetic component is smaller than the enthalpy component, which agrees
with the results from the analytical thermal wind model in
Sect. . The ratio of kinetic to enthalpy separation-squared
is about 0.37 on average.
Insight from Eq. () shows that it is because Earth is a rapidly
rotating planet (i.e. Ωe is large) resulting in smaller
geostrophically balanced flow for the same equator–pole temperature
difference. The possibility to reveal this reason stems from the separation
metric being induced from the energy norm which respects the fundamental
dynamical ratios in the system. Such dynamical reasoning would not be
possible if the metric was arbitrarily constructed with user-prescribed
ratios.
It has been shown in the Lorenz energy cycle that
baroclinic eddies are the primary driving force for the energy exchange in
mid-latitudes. However, the eddies are neglected in the calculation above with
zonal mean data. To reveal the contribution to the separation metric from
eddies, the same separation-squared is calculated from the full 3-D data as
shown in Fig. b. The kinetic separation-squared is found to be
comparable to the enthalpy separation-squared when the mid-latitude eddies
are included. We attribute the difference between Fig. a and b to
the existence of mid-latitude eddies. So on average, eddies contribute 78 %
to the total kinetic separation-squared and 28 % to the total enthalpy
separation-squared.
(a) Kinetic (red), enthalpy (green) and surface pressure
(blue) separation-squared between CFSR monthly mean data and its annual mean
climatology in the tropical region (25∘ S–25∘ N) up to
100 mb. (b) Time–latitude cross-section of the logarithm to base
10 of the total separation-squared integrated zonally and vertically up to
100 mb in the tropical region.
(a) Kinetic (red), enthalpy (green) and surface pressure
(blue) components of separation-squared between CFSR monthly mean data and
its annual mean climatology in the equatorial tropics
(5∘ S–5∘ N) integrated over the whole atmosphere column.
(b) ‖QBO index‖ (black) and RC(4, 5) of kinetic
separation-squared (magenta) shifted backward by 5 months. Black solid
(dashed) lines represent westerly (easterly) phase of the QBO.
(c) Lag correlation of ‖QBO index‖ with the RC(4, 5) of
kinetic separation-squared. (d) Lag correlation of ‖QBO index‖
with kinetic separation-squared at different pressure levels. The black
contours denote the 99 %-confidence level. In (c, d), a positive
lag denotes the signal leading the QBO index.
Multi-variate phase space for a hydrostatic flow in pressure
coordinate. For ease of visualization, wind u is shown as a 1-D axis
instead of a 2-D plane and each axis represents one among an infinite number of
degrees of freedom on the surface (for pH) or in the volume
(for T and u) of a fluid. The length of phase vectors A and
B are a and b respectively while their separation is c in this
subspace. O is the null vector. The spheres represent hyper-surfaces
of constant energy.
These percentages are consistent with the contribution from eddies in the
Lorenz energy cycle, where eddies contribute 71 % to the total kinetic
energy and 31.8 % to the total APE .
The enthalpy separation-squared is not conceptually related to Lorenz's
definition of APE in Eq. (). There are superficial resemblances
because of the quadratic form, but in Eq. (), no fixed reference
state is assumed and the atmospheric lapse rate plays no role. Nonetheless,
the partition between zonal mean and eddy contributions in enthalpy
separation-squared and in APE are comparable because Γ is nearly
constant in the troposphere while climatological temperature Tclim∼T¯, leading to enthalpy separation-squared from the annual mean
climatology being roughly proportional to APE.
The coefficient of temperature difference δT2 is
different in our linearized separation-squared in Eq. () from that
in the often used metric MT81 in Eq. ()
. The enthalpy separation-squared
is recalculated using progressively modified formulae in Fig. c
(see the details in the caption). It is found that using constant reference
temperature To=270 K and constant reference pressure po=1000 mb does not change the enthalpy separation-squared much.
But the increase of the coefficient by a factor of 2 from our metric to
MT81 (green to grey line in c) makes the enthalpy
contribution to the total separation-squared twice as significant!
Although the surface pressure separation-squared is 1 order of magnitude
smaller than the enthalpy separation-squared (Fig. b), the
coefficient of δpH2 also differs between our
metric and MT81. Figure a and b compare the northern
mid-latitude terrain and the topographic contribution to the surface pressure
separation-squared. However, the boundary-layer enthalpy contribution to the
surface pressure separation-squared is nearly an order of magnitude larger
and maximizes over the central Pacific Ocean (Fig. c). If
MT81 was used instead, the surface pressure separation-squared
would be considerably smaller (see Fig. c and d).
Tropical oscillations
The separation-squared between CFSR monthly mean 3-D data and its annual mean
climatology in the tropical region (25∘S–25∘ N) up to
100 mb is calculated and the results between 2001 and 2009 are shown in
Fig. a. Surface pressure separation-squared, like kinetic and
enthalpy separations-squared, shows a synchronous semi-annual oscillation as
oceans cover more area than land in the tropics.
The kinetic separation-squared is about 1 order of magnitude larger than
the enthalpy separation-squared, quite unlike in the mid-latitudes (see
Figs. a and b). This can be explained by the near
constancy of temperature and surface pressure as opposed to large seasonality
of wind in the tropics, e.g. due to the monsoons, whereas geostrophic and
thermal wind balance necessitate surface pressure and temperature to have
accompanying large variation to wind variation.
The semi-annual oscillation at different latitudes is further investigated in
Fig. b. Contributions from the higher tropical latitudes are an
order of magnitude larger than from the equatorial latitudes, which attests
to the constant climate near the equator. The seasonality is stronger in the
Northern Hemisphere compared to the same latitude in the Southern Hemisphere.
We next focus on the equatorial atmosphere to see what the separation metric
can reveal about tropical dynamics. The separation-squared between CFSR
monthly mean data and its annual mean climatology in the equatorial tropics
(5∘S–5∘ N) integrated from the surface to the stratopause
(1 mb) is shown in Fig. a. Further investigation of the
separation-squared level by level shows that on average, the kinetic and
enthalpy separations-squared in the stratosphere (70 to 1 mb) contribute
39.3 and 47.3 % respectively to the kinetic and enthalpy
separations-squared in the whole column.
This is noteworthy because the stratosphere only makes up about 10 % of
the atmospheric mass but it accounts for up to about 40 % of the combined
monthly variance of the equatorial troposphere and stratosphere as measured
by the energy norm-induced metric.
The quasi-biennial oscillation (QBO) is a quasi-periodic reversal of the mean
zonal wind in the equatorial stratosphere, and is well-known to influence the
global stratosphere through modulation of zonal wind, temperature, humidity
and the meridional circulation . The meridional
distribution of QBO amplitude is approximately Gaussian, centred at the
equator with a 12∘ half-width .
Although the QBO has a signature in temperature, it is weak because
geostrophic balance is not dominant near the equator and thus the QBO signal
is not identifiable in the enthalpy separation-squared against large signals
arising from seasonal variation in insolation. We present only the analysis
of the kinetic separation-squared here, which is a very good approximation to
the total separation-squared because of its overwhelmingly large
contribution.
Using singular spectrum analysis (SSA), we first decompose the kinetic
separation-squared into a trend, seasonal oscillation and interannual
oscillation, leaving a residue. Between 1979 and 2009, there are 372 sample
points in time. The window length for SSA used is 36 sample points, i.e.
3 years. The first reconstructed component, RC(1), explains 90.9 % of the
total variance. It traces the decadal variation which is in anti-phase to the
10.7 cm solar flux and has a secular rising trend.
The semi-annual oscillation due to the seasonality of hemispheric insolation
is captured by RC(2, 3), explaining 3.6 % of the total variance. RC(4, 5)
explains 1.4 % of the total variance and when shifted backward by 5 months
matches the QBO index very well (Fig. b), including years when the
QBO period is longer than average, e.g. 1999 to 2002. (The QBO index is
defined as the zonal wind at 30 mb over Singapore.) RC(4, 5) is lag
correlated with the absolute value of the QBO index in Fig. c. The
maximum correlation score of 0.53 is attained when the RC(4, 5) lags behind
the QBO index by 5 months. The correlation score peaks every 28 months which
is the average period of the QBO.
The lag correlation of the absolute value of the QBO index with the (full)
kinetic separation-squared at different pressure levels is shown in
Fig. d. The correlation score maximizes at 30 mb with zero lag
simply because the QBO index is defined at this level. Since the QBO phase
propagates downward, the kinetic separation-squared leads the QBO index at
higher pressure levels than 30 mb and lags behind at pressures lower than
30 mb. Only the tilted positive correlation band centred at zero lag
denotes a real physical connection. The other tilted correlation bands
located at about multiples of 7 months away are just mirrored images produced
by the quasi-periodicity of the QBO. The correlation score drops rapidly in
the troposphere and becomes insignificant at the 99 % confidence level
(except around 500 mb). To minimize the effect of auto-correlation in time,
we assume that the number of independent samples is 124, which is the
number of seasons in our time series, giving the number of degrees of freedom
as 122.
There is some indication that QBO has a significant but weak influence in the
mid-troposphere around 500 mb. Such an influence may not be as unreasonable
as it first sounds because of the disproportionately large stratospheric
contribution to equatorial atmospheric variance mentioned earlier. A
plausible dynamical reason could be that the zonal mean wind in the lowermost
stratosphere (∼70 mb) modifies the vertical propagation of equatorial
waves, reflecting certain waves downwards so that their trapped energy
maximizes in the mid-troposphere at the peaks of QBO easterly or westerly
phases. Because we use a metric induced by the energy norm, when the energy
of the atmospheric state is enhanced, the separation of that state from the
annual mean climatology is correspondingly enhanced.
Discussion and summary
To date, much of the literature's rationale to the definition of a metric
runs along two lines of thinking which are not mutually exclusive:
to employ the quadratic form of a norm and its induced metric beyond
the restricted dynamical regime for which the norm is proven to obey a
conservation law , with the confidence that the form
is at least valid in that regime;
to justify the quadratic form of a metric based on its simplicity
and on dimensional consistency among the contributions by
different state variables: the weighing coefficient on each variable depends
on the suitable choice of a convenient reference state, certain physical
constants and dimensionless numbers, as well as the practical importance of
emphasizing that variable.
Neither line of thinking is without its merits and both arguments are
substantial enough if practical application demands utility more than
theoretical rigour. For instance, to find a singular vector or conditional
nonlinear optimal perturbation (CNOP) in a model forecast for adaptive
observation , knowing that temperature has larger
normalized error variance than wind would favour a metric
definition that emphasizes temperature deviations more, such as
MT81 instead of Eq. (). In that case, extending the
use of a metric beyond the regime for which it was originally designed –
MT81 was formulated by for linearized,
adiabatic, inviscid flows and is constant for the forecast of such flows in
between consecutive data assimilations – is justifiable at least because
the practical use of the metric in nonlinear, forced-dissipative regimes
enables important advancement in NWP.
There are other practical considerations: Sect. 4a of
mentions the relevance of numerical discretization schemes in determining
whether a norm is practically invariant or not. Sections 4 and 6 of
distinguish the analysis error covariance metric for
practical predictions, which depends on the observation network and the data
assimilation scheme, against the geophysical fluid dynamics (GFD) covariance
metric for GFD studies, which depends on dynamics only and is defined from
the invariant measure associated with the system's attractor. The results of
ensemble predictions, error growth analysis and predictability studies will
be sensitive to the choice of the metric. Section 5 of
showed that the MT81 metric may be more suitable for practical
prediction studies than in pure GFD problems on predictability.
Having recognized the merits of the above approaches, it is instructive to
examine the comparative advantages of our theoretical approach. For that, we
need to elucidate the underlying physical basis of the norm-induced metric.
Figure is a Cartesian representation of the phase space
(u,T,pH) constructed for a hydrostatic flow in
pressure coordinate. To our knowledge, this is the first time that quantities
like T and pH are constructed to serve as phase
coordinates and we emphasize that only in these constructed coordinates does
the square-root of true energy satisfy all the axioms of a norm.
Emod of Eq. () is the Euclidean norm on the
vector space of (u,T,pH); it is not even a
norm in the conventional vector space of (u,T,pH) because it
does not have absolute homogeneity there.
With reference to Fig. , Emod is used to
measure a and b, the lengths of phase vectors A and B respectively. The M
of Eq. () is the metric used to measure the separation c between
A and B. As the metric is induced by the norm, the separation of a phase
vector from the null vector O is identical to the phase vector's
length. This means that a, b and c are measured with the same “ruler”. By
using any other metric in a normed vector space, we are measuring c on a
different ruler from a and b, which is admissible mathematically but goes
against physical sense.
For adiabatic, inviscid flows, it is a law of physics that energy E is
conserved and hence a and b are invariant. The vectors A and B move on
constant energy hyper-surfaces which take the form of hyper-spheres when the
phase coordinates (u,T,pH) are scaled by
factors of dp/2g, cpdp/g,
(cpTr+ΦH)/g respectively and have
common units of m-1. Unless the flow is linear (like the case in
), c will generally not be invariant. Then the norm
induced metric can indeed be used to detect changing separations between A
and B while the invariant lengths of A and B provide physical justification
for using the same “ruler” to measure separation. In this way, we avoid the
inherent contradiction that MT81 faces: it is only useful when
the norm is not conserved (see the end of Sect. ).
Moreover, linearization of the flow is never required for the conservation of
energy. Without detracting from the last statement, where mathematically
valid, the separation metric can be transformed into the tangent linear space
of conventional but non-Cartesian coordinates (u,T,pH) where
either A or B provides a realistic nonlinearly evolving reference state for
the linearization of the metric. In these coordinates, the norm and metric
take on a non-Euclidean form.
There is another advantage of measuring a, b and c on the same
“ruler”: the angle ψ between vectors A and B in Cartesian coordinates
(u,T,pH) can be consistently defined by the
cosine rule:
cosψ=a2+b2-c22ab,
which is equivalent to the definition of the inner product:
〈A,B〉=a2+b2-c2/2.
For the metric in Eq. (), the inner product defined in this way is
〈A,B〉=1g∫A∫0pr12uA⋅uB+cpTATBdpdA+1g∫AcpTr+ΦHpHApHBdA,
where the superscripts on the variables refer correspondingly to states A
and B. Equation () may be contrasted against Eq. (11)
of which is the inner product in
(u,T′,ps′)-space related to the
MT81 metric, reproduced in the notation of this paper as
〈A,B〉P98=12g∫A∫0pouA⋅uB+cpToT′AT′BdpdA+12g∫ARTopolnpsApolnpsBpodA.
The set of angles a phase vector makes with the Cartesian axes fixes the
direction of the phase vector. The notions of direction and inner product are
fundamental to many concepts and applications in predictability (e.g.
Lyapunov exponents) and optimization of error growth (e.g. CNOP). Like our
definition of separation, our definitions of direction and inner product
ultimately rest upon the physical principle of energy conservation as the
basis for the invariance of the norm and Euclidean geometry is manifest in
(u,T,pH)-space. In contrast, the set of metric
MT81, norm ET81 and inner product
〈A,B〉P98 respects Euclidean geometry in
(u,T′,ps′)-space, but ET81 is
not energy, causing the norm to vary in nonlinear, adiabatic, inviscid flows.
In such flows, quantifying the separation and angle between two state
vectors by MT81 and 〈A,B〉P98 would
be like measuring distance and angle with elastic rulers and protractors.
Appendix A delves further into the origin of the difference between
ET81 and energy E.
Placed in the context of applications like ensemble forecast, the above
theoretical development provides a physically based metric that can be used
to, for instance, measure the spread of member states about the observation
in multi-variate phase space where no single variable can summarize the model
performance, such as at the surface or in the tropics
. The development of such multi-variate spread diagnostics
would complement existing univariate spread measures .
Another use can be in error growth analysis to define the norm of a CNOP used
to identify area targets for observation . A major practical
advantage in the above applications would be that even large separations can
be rigorously quantified using the Cartesian coordinates in which the energy
norm is Euclidean, such as illustrated in Fig. a of
Sect. . This would be essential, for example, in
shallow-water simulations of tsunamis .
The importance of the non-Euclidean form of the metric for nonlinear flows
illustrated in Sect. (Fig. ) is an important
advancement in the definition of separation. For highly nonlinear,
non-hydrostatic atmospheric flows at mesoscales, especially those manifesting
strong convection such as around the core region of a tropical cyclone (TC),
the separation between atmospheric states involves a kinetic–buoyant energy
inter-conversion term, δwδρ in Eq. (). The
practical implementation of such a metric for computing CNOP may result in a
discernible impact on the areas targeted for observation to improve TC
intensity forecasts.
By agreeing on the invariant norm relevant to the dynamics of a system, for
instance the energy norm in pressure coordinate, the relative ratios of
contribution among the conventional state variables, δu, δT and δpH in this case, to the total separation-squared are
no longer arbitrary to some dimensionless constants or dependent on the
user's choice of the reference state, as the case would be for a metric
constructed from dimensional consistency arguments alone.
For example, the small ratio of enthalpy to kinetic components of
separation-squared in the tropics (Fig. a) compared with the
mid-latitudes (Fig. a, b) cannot be increased in an ad hoc manner,
e.g. by replacing the nonlinear reference-state T by the climatological
amplitude of diurnal temperature fluctuations in Eq (). The reason
is that this ratio is reflective of the lack of geostrophic balance in the
monthly mean tropical climate (i.e. contrary to the case in Eq. ()
and Fig. ). Likewise, in the mid-latitudes, the use of
MT81, where the enthalpy component is roughly doubled
(Fig. c), would not be recommended if the dimensionless ratio of
system constants governing the dynamics of thermal wind balance in
Eq. () is to be respected. Nonetheless, we do not believe having
doubled the enthalpy contribution in MT81 would detract from the
qualitative conclusions of much previous work even if details might have been
altered, e.g. the consistency of the “energy” norm to the “analysis error
covariance metric” in .
We have given a firm theoretical basis for the contribution of surface
topography on the metric. Previous theoretical literature (e.g.
) did not allow for the presence of topography. It would
not be possible to guess the form of topographic influence by dimensional
analysis alone. For example, by dimensional analysis, topography could well
modify the enthalpy contribution as cpTo+ΦHδT/To2 in Eq. () instead
of modifying the surface pressure contribution in Eq. (),
especially looking at the form of Eq. (). With our approach, the
topographic term is negligible in pressure coordinate (Fig. b, c)
because ΦH/(cpT)≲1%, and this is also true in
isentropic coordinate, see Eq. (). But one would not be able to
consistently neglect the topographic effect if one used the expression
cpTo+ΦHδT/To2 because surface pressure separation-squared can be about 1%
of enthalpy separation-squared (e.g. in Fig. a) and is retained
within the metric expression. In height coordinate, topography does not even
appear except as the fixed lower limit of vertical integration for the
semi-infinite atmosphere.
By using the energy norm-induced metric, we detected a weak but statistically
significant teleconnection between the QBO phase in the lower stratosphere
and the monthly variability at mid-tropospheric levels (Fig. d)
which may be worth further investigation in future. At this early juncture,
it is understandable if a teleconnection is selectively picked out by our
metric for atmospheric variation because of the principle of energy
conservation on which the metric is based: a longer state vector due to
accumulation of tropospheric energy by equatorial wave reflection from the
lower stratosphere would manifest larger variation from the climatic mean
state since we use the same “ruler” to measure energy and separation in
phase space (see Fig. ). If the enthalpy contribution
cpT(δT/T)2 was artificially exaggerated roughly a
hundredfold by normalizing the temperature difference by its variance
ΔT instead of by T, i.e. cpT(δT/ΔT)2, the
metric would now be dominated by temperature variability in which the QBO
signal is swamped by seasonal signals. The above teleconnection between the
mid-troposphere and the QBO index would be lost when using such an ad hoc
metric.
While useful, the energy norm is not the only invariant norm from which a
metric can be induced. Other dynamical invariants, e.g. enstrophy
and wave activity , could be used. One
only needs to construct the phase space judiciously following the approaches
demonstrated in Sects. and so that
the invariant quantity takes the form of a Euclidean norm. Hence, the above
theoretical advantages may potentially be relevant to most problems with a
conserved physical quantity. For instance, in homogeneous, isotropic
turbulence of a 2D incompressible fluid, the metric induced by the enstrophy
norm may be useful in investigating chaotic dynamics of turbulence: e.g.
defining ζ as absolute vorticity, one might consider the spectral power
of ∫Aδζ2dA in the enstrophy cascading
inertial sub-range as a measure of the separation between two mature
turbulent flows.
In summary, we propose a new two-step approach in tackling the problem of
metric definition: (1) constructing the phase space specifically so that an
invariant based on a physical conservation law is the Euclidean norm
on this space; and (2) defining the norm-induced metric to quantify the
separation in phase space between two states. This methodology is
mathematically rigorous and physically meaningful. The norm can be invariant
even for nonlinear flows and the norm-induced metric is valid even for large
separations. We have applied this approach to examine analytical examples and
realistic reanalysis data, and discussed its potential applications in
ensemble prediction, error growth analysis and predictability studies. But we
note that practical and other theoretical considerations may favour
alternative approaches to defining the metric. Finally, the separation metric
in this study is developed for dry atmospheres. We are working next on the
separation metric including moisture.