• Bishop, C. H., 1993: On the behaviour of baroclinic waves undergoing horizontal deformation. II: Error-bound amplification and Rossby wave diagnostics. Quart. J. Roy. Meteor. Soc.,119, 241–267.

  • Bouttier, F., 1994: Sur la prévision de la qualité des prévisions météorologiques. Ph.D. thesis, Université P. Sabatier-Toulouse III, 240 pp.

  • Bretherton, F. P., 1966: Baroclinic instability and the short wave cutoff in terms of potential vorticity. Quart. J. Roy. Meteor. Soc.,92, 335–345.

  • Davies, H., and C. Bishop, 1994: Eady edge waves and rapid development. J. Atmos. Sci.,51, 1930–1946.

  • Eady, E., 1949: Long waves and cyclone waves. Tellus,1, 33–52.

  • Farrell, B. F., 1982: The initial growth of disturbances in a baroclinic flow. J. Atmos. Sci.,39, 1663–1686.

  • ——, 1984: Modal and non-modal baroclinic waves. J. Atmos. Sci.,41, 668–673.

  • ——, 1988: Optimal excitation of neutral Rossby waves. J. Atmos. Sci.,45, 163–172.

  • Ghil, M., 1989: Meteorological data assimilation for oceanographers. Part I: Description and theoretical framework. Dyn. Atmos. Oceans,13, 171–218.

  • Gill, A., 1982: Atmosphere–Ocean Dynamics. International Geophysics Series, Vol. 30, Academic Press, 662 pp.

  • Held, I. M., 1985: Pseudomomentum and the orthogonality of modes in shear flows. J. Atmos. Sci.,42, 2280–2288.

  • Holton, J. R., 1979: An Introduction to Dynamic Meteorology. International Geophysics Series, Vol. 23, Academic Press, 391 pp.

  • Houtekamer, P., 1995: The construction of optimal perturbations. Mon. Wea. Rev.,123, 2888–2898.

  • Joly, A., 1995: The stability of steady fronts and the adjoint method:Nonmodal frontal waves. J. Atmos. Sci.,52, 3082–3108.

  • ——, and A. J. Thorpe, 1991: The stability of time-dependent flows:An application to fronts in developing baroclinic waves. J. Atmos. Sci.,48, 163–182.

  • Kalman, R., 1960: A new approach to linear filtering and prediction problems. Trans. ASME,82D, 35–45.

  • ——, and R. Bucy, 1961: New results in linear filtering and prediction theory. Trans. ASME,83D, 95–108.

  • Mukougawa, H., and, T. Ikeda, 1994: Optimal excitation of baroclinic waves in the Eady model. J. Meteor. Soc. Japan,72, 499–513.

  • O’Brien, E., 1992: Optimal growth rates in the quasigeostrophic initial value problem. J. Atmos. Sci.,49, 1557–1570.

  • Rotunno, R., and M. Fantini, 1989: Petterssen’s type B cyclogenesis in terms of discrete, neutral Eady modes. J. Atmos. Sci.,46, 3599–3604.

  • Snyder, C., 1996: Summary of an informal workshop on adaptive observations and FASTEX. Bull. Amer. Meteor. Soc.,77, 953–961.

  • View in gallery

    Linear amplifications for the reference inner product 〈 · , · 〉0 versus wavenumber: normal mode (solid), optimal singular mode μ1 (dashed), and damped singular mode μ2 (dashed–dotted). Different timescales are shown: (a) T = 6 h, (b) T = 24 h, (c) T = 48 h, (d) T = 72 h. In (a), arrows indicate the location of particular wavenumbers: most amplified normal mode ωn, most amplified singular mode ωs, and smallest neutral normal mode ωc.

  • View in gallery

    Linear amplifications for the potential energy inner product 〈 · , · 〉A versus wavenumber: normal mode (solid), optimal singular mode μ1 (dashed), and damped singular mode μ2 (dashed–dotted). Different timescales are shown: (a) T = 6 h and (b) T = 24 h.

  • View in gallery

    Structure of the SM1 and NM geostrophic meridional wind υg: normal mode NM (left) and singular mode SM1 for the reference inner product 〈 · , · 〉0 (right). The timescale is T = 6 h. Note also that the neutral NM is the one with maximum intensity at the surface.

  • View in gallery

    Structure of the SM1 geostrophic meridional wind υg: the singular mode SM1 is for the reference inner product 〈 · , · 〉0. The timescale is T = 48 h.

  • View in gallery

    Structure of the SM1 geostrophic meridional wind υg: the singular mode SM1 is for the potential energy inner product 〈 · , · 〉A. The timescale is T = 6 h.

  • View in gallery

    Structure of the SM1 geostrophic meridional wind υg: the singular mode SM1 is for the potential energy inner product 〈 · , · 〉A. The timescale is T = 48 h.

  • View in gallery

    (a), (c) Variances of geopotential in physical space at initial time, and (b), (d) at day 1 for a (a), (b) shortwave and a (c), (d) longwave. Values are in m4 s−4. Maximum values are also indicated (arrows).

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Linear Amplification and Error Growth in the 2D Eady Problem with Uniform Potential Vorticity

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  • 1 Météo-France, Centre National de Recherches Météorologiques/Groupe de Météorologie à Moyenne Echelle, Toulouse, France
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Abstract

The concept of a singular mode underlies optimal linear amplification theories. This concept is studied in the frame of the two-dimensional, quasigeostrophic Eady problem with uniform potential vorticity. Analytical solutions are produced for the relevant physical norms. Exact relations are also derived for the amplifications, which give the lower and upper bounds to any linear development. Results show significant differences in the structure of the singular modes, as well as in the associated amplifications, when the horizontal wavenumber is varied or the inner product is changed. It is found that the singular modes can depart significantly from the normal modes, though the dynamics of the problem are very simple. Comparisons with previous works are also performed. Finally, the derived equations are used to present the linear evolution of error growth within the Eady problem, as predicted by a Kalman filter. Considerations on the spectral space error covariance matrix are made, and a particular case of error dynamics in the 2D physical space is shown. The derivation of the general algebraic solutions is included in the .

Corresponding author address: Dr. Claude Fischer, Météo-France, CNRM/GMME, 42, Avenue Gustave Coriolis, 31057 Toulouse, Cedex, France.

Email: Claude.Fischer@meteo.fr

Abstract

The concept of a singular mode underlies optimal linear amplification theories. This concept is studied in the frame of the two-dimensional, quasigeostrophic Eady problem with uniform potential vorticity. Analytical solutions are produced for the relevant physical norms. Exact relations are also derived for the amplifications, which give the lower and upper bounds to any linear development. Results show significant differences in the structure of the singular modes, as well as in the associated amplifications, when the horizontal wavenumber is varied or the inner product is changed. It is found that the singular modes can depart significantly from the normal modes, though the dynamics of the problem are very simple. Comparisons with previous works are also performed. Finally, the derived equations are used to present the linear evolution of error growth within the Eady problem, as predicted by a Kalman filter. Considerations on the spectral space error covariance matrix are made, and a particular case of error dynamics in the 2D physical space is shown. The derivation of the general algebraic solutions is included in the .

Corresponding author address: Dr. Claude Fischer, Météo-France, CNRM/GMME, 42, Avenue Gustave Coriolis, 31057 Toulouse, Cedex, France.

Email: Claude.Fischer@meteo.fr

1. Introduction

Traditional developments in analytical dynamics of the atmosphere extensively use linear instability theories. At midlatitudes for example, planetary waves and synoptic-scale cyclones are explained by mixed inertia–Rossby waves and barotropic/baroclinic instabilities, respectively. Both types of structures are found as solutions x to the following eigenvalue problem:
i1520-0469-55-22-3363-e1
where L is the set of linearized equations governing the flow and x is a “normal mode” of L. If x0 is the initial value of the normal mode, then it is straightforward that for each time t: x(t) = x0eσt. The normal mode therefore never changes its initial shape in space. The success of these approaches is due to their ability to predict correct growth rates and phase speeds when compared to observation and climatology. However, it was long recognized that normal modes are not the most rapidly growing structures in geophysical flows. Also, the constraint of a stationary spatial structure is very restricting. Therefore, initial value approaches have been studied recently. In this alternative approach, the linear growth of a perturbation is measured by its ability to extract a large amount of energy from the basic state within a finite time. For example, the original idea of Farrell (1982) is to introduce a continuous set of solutions at initial time to obtain a combination of normal modes whose total energy grows quicker than that of any normal mode separately. Typically, solutions in this continuous set are not space–time separable in contrast to any individual solution from the discrete set, which is separable. Only when both types of solutions are considered is the completeness of the set of solutions achieved in a nonuniform potential vorticity flow. For instance, in the Eady problem, Farrell (1984) exhibits linear solutions that are obtained as the sum of a separable function and of a plane wave, nonseparable solution. The initial value problem, as set forth by Farrell, then consists in finding the best initial state so that the two solutions interact constructively with the basic state and large amounts of total energy can be deposited in the separable structure. By doing so, Farrell shows that development is possible, even for the range of neutral normal modes. Farrell’s idea is revisited by Rotunno and Fantini (1989). In their study, a linear combination of three-dimensional neutral modes is constructed so that the initial surface pressure is uniform. Because the neutral modes are not orthogonal in the sense of physical inner products (e.g., kinetic energy), they can interact and extract some energylike quantity from the basic state (Held 1985). Therefore, the linear perturbation amplifies into a deepening surface depression at a finite time T. Note that Rotunno and Fantini use only separable structures because they impose the perturbation potential vorticity to be uniform, a case that filters the continuous set of solutions.

Davies and Bishop (1994) study the maximum instantaneous growth of the potential temperature in the Eady problem with uniform interior potential vorticity. They show that maximum growth at the surface and tropopause boundaries is achieved when the surface wave is lagging behind the upper wave, with a −π/2 phase shift and with equal amplitudes at both boundaries. If a finite time interval T is considered for the wave’s development in the Davies and Bishop study, then the optimal wave structure is the one that changes its phase tilt symmetrically, from the initial tilt to quadrature over [0, T/2], and from quadrature to the normal-mode asymptotic tilt over [T/2, T]. Bishop (1993) also studies baroclinic, uniform potential vorticity waves that have maximum growth rates over a prescribed time in an evolving basic state. Following Bretherton (1966), he stresses that the optimal wave should keep a phase tilt between its surface and tropopause anomalies of −π/2 as long as possible. Eventually, the introduction of a strain deformation reduces optimal growth, mainly through the change of the horizontal wavenumber, which decreases the Rossby height.

Joly (1995) reformulates the initial value problem for optimal development. Three points are found to be important:

  • the time T over which development occurs;
  • the development, which is measured by computing a quadratic function q(x) of the perturbation x (energy, enstrophy, etc.); and
  • the resolvent system of equations, introduced earlier by Joly and Thorpe (1991). It consists of the time integral of the tangent linear system L:
    i1520-0469-55-22-3363-e2

For convenience, the resolvent set is abbreviated as M(0, T) = M. An inner product is chosen in the space of the model solutions. The inner product will hereafter be noted with angle brackets. It measures the quadratic function defined previously:
xxqx
For analytical studies, it is suitable to take positive definite products. The initial value problem as formulated by Joly (1995) then becomes: find x0 such that the time integral x(T) has a maximum amplification α over the interval of time T. The amplification can then be written as
i1520-0469-55-22-3363-e4
with M* standing for the adjoint operator of M with respect to the inner product 〈 · , · 〉. Each eigenvector of M*M is a singular mode of the problem. The optimization problem is then equivalent to finding the eigenvector x1 with the greatest eigenvalue μ1. Farrell (1988) uses the very same definition for his optimally growing modes, without referring explicitly to “singular vectors” (see his section 4). Joly (1995) displays numerical solutions of singular modes for several norms within a uniform potential vorticity flow. It is of interest that the shapes of the solutions depend on the norm, so that the singular vectors are found to be either baroclinic or barotropic modes. In Joly’s study, the time T is set somewhat arbitrarily to 48 h by comparison with typical timescales for secondary cyclogenesis in the atmosphere.

Mukougawa and Ikeda (1994) as well as Farrell (1988) exhibit singular modes for the Eady problem with nonuniform potential vorticity. In both studies, the singular modes can have a fine, leafletlike structure in the domain interior, with almost no significant gradients at the boundaries. Such patterns are typical of flows with nonuniform interior potential vorticity.

In this paper, we investigate the problem of singular modes with analytical tools. The 2D quasigeostrophic Eady problem with uniform potential vorticity is used because its simple formulation allows for tractable calculations. By uniform potential vorticity, it is understood that the basic state has a constant positive value, while the perturbations have zero potential vorticity. Insight is given into the structures of the singular modes, and comparisons with the normal modes are made. Different inner products are considered in order to stress the dependence of the singular vectors on the choice of the physical norm. The study is proposed as a guideline for comparisons with numerical results that are under way with numerical, balanced models. As is shown in section 2, and due to the requirement of uniform potential vorticity, the singular vectors result from the nonorthogonality of each couple of upper and lower waves, so that the problem is two-dimensional.

The prediction of forecast error growth is visited in the second part of this paper. Eady (1949) was already concerned with the impact of unstable modes on the growth of initial errors. In particular, he makes the point that the probability of any single linear development is“unequally distributed,” a property that is expressed here by the error covariance matrix. The algebraic tools for error growth calculations are similar to those of the singular mode research. Advantage of this similarity is taken and a discussion of error covariance dynamics in the Eady problem is performed. The particular case presented here merely represents the time evolution of error variances in a Kalman filter under the perfect model assumption (Ghil 1989) after an initial step of data assimilation that provided locally zero variance.

Section 2 deals with the study of the singular modes. The basic parameters of the Eady problem are recalled in section 2a; inner products are presented in section 2b; the operators L and M are presented in section 2c. General solutions for the singular vectors in a 2D problem are presented in the appendix. The singular modes are exhibited and described in section 2d, along with the amplifications. Section 3 shows a particular case of linear error growth. After some definitions (in section 3a), some properties of spectral error covariances are presented in section 3b and the error variances in the physical X–Z vertical plane are discussed in section 3c.

2. Singular modes

a. Basic equations

In this section, the two-dimensional Eady problem is recalled briefly and some general considerations concerning the desired solutions are addressed. The atmosphere is taken to be a two-dimensional fluid in an X–Z coordinate system with lower and upper rigid lids at Z = −H and Z = +H, respectively. Free-slip boundary conditions apply. The flow is in hydrostatic equilibrium and it is separated into a large-scale, geostrophic part and a perturbation. The large-scale flow U is driven by a transverse thermal gradient so that
i1520-0469-55-22-3363-e5
where uppercase letters stand for large-scale fields. All other Y-derivatives are assumed to be zero. The basic physical parameters bear their usual significance: f is the Coriolis parameter, N the square root of the Brunt–Väisälä frequency, g the acceleration of gravity, θ the potential temperature, and Φ the geopotential height. It is also convenient to define the following parameters (see Gill 1982):
  1. Λ = dU/dZ so that Λ−1 is a characteristic timescale;
  2. κh = 2π/L, the horizontal wavenumber;
  3. HR = f/h, the Rossby height.(6)
First, we recall the solutions of the Eady problem as presented in Gill (1982). The quasigeostrophic assumptions are retained, and the geopotential perturbation Φ verifies the following linear, balance condition:
i1520-0469-55-22-3363-e7
in the interior domain. Thus, solutions for Φ are in the general form
i1520-0469-55-22-3363-e8
The requirement for zero vertical velocity at the boundaries leads to the following tendency equations for the functions A(X, t) and B(X, t):
i1520-0469-55-22-3363-e9
Periodic solutions for (9) are sought:
i1520-0469-55-22-3363-e10
and such solutions exist provided that
i1520-0469-55-22-3363-e11

The latter equation contains the information concerning the phase speed and the growth rate of the linear modes, provided that c = cr + ici.

For the subsequent developments, it is sufficient to fix a horizontal channel length L, and to study the harmonic solutions that form a countable set of wavenumbers. Then, solutions read
i1520-0469-55-22-3363-e12
In the following, we use the couple of basis functions (αk, βk):
i1520-0469-55-22-3363-e13
and we call Ek the two-dimensional vector space of all linear combinations of αk and βk for fixed k and E the vector space for all k.

b. Inner products

1) General approach

The choice of the inner product in the vector space E is crucial. Indeed, algebraically, the adjoint of any operator f is defined with respect to a predefined inner product
xyExfyfxy

Thus, the singular modes also depend on this choice.

In this study, we choose a reference inner product derived from the norm:
i1520-0469-55-22-3363-e15
where f stands for the complex-conjugate of f. Note that
i1520-0469-55-22-3363-e16
and that
i1520-0469-55-22-3363-e17
Therefore, if we define now
i1520-0469-55-22-3363-e18
then the couple (χk,ψk) forms an orthonormal basis of Ek for the inner product 〈 · , · 〉0. The results above also show that all normal modes built with the (αk, βk)kN° set will be orthogonal two by two if they do not have the same horizontal wavelength. Therefore, there is no need for considering all the harmonics and in the following; we only study one fixed wavelength 2π/κhk = L/k and restrict ourselves to one two-dimensional subspace Ek at a time. For convenience, we set k = 1. Note that if several harmonics are considered, then the total amplification α is the sum of the elementary amplifications at each wavenumber, and local effects may be obtained by interferences. As in Joly (1995), we also define physical inner products, in order to link the singular modes with the physics included in the Eady problem, beginning with potential energy.

2) Potential energy

Potential energy is defined for the perturbation by the positive-definite quantity
i1520-0469-55-22-3363-e19
in the Boussinesq form. Since the perturbation is in hydrostatic equilibrium,
i1520-0469-55-22-3363-e20
so that the inner product 〈 · , · 〉A that defines potential energy in Ek reads
i1520-0469-55-22-3363-e21
Let (ak, bk) be the two coefficients of ϕ in the orthonormal basis (χk, ψk); then the potential energy of the perturbation reads
i1520-0469-55-22-3363-e22
Here, we have used the domain integrals of χk and ψk as defined with the inner product 〈 · , · 〉0. We deduce from this result that the following matrix,
i1520-0469-55-22-3363-e23
represents the 〈 · , · 〉A inner-product matrix in the (χk, ψk) basis. Thus, an alternative form for A reads
i1520-0469-55-22-3363-e24

3) Kinetic energy

Perturbation kinetic energy is defined as
i1520-0469-55-22-3363-e25
with
i1520-0469-55-22-3363-e26
The same development as for 〈 · , · 〉A leads now to the following inner-product matrix SK:
i1520-0469-55-22-3363-e27
with I the identity matrix. This expression implies simply a renormalization by a real scalar. In other words, it shows that, for the 2D Eady problem, the inner products 〈 · , · 〉K and 〈 · , · 〉0 are equivalent. In particular, adjoint operators and singular modes are the same for both products [see Eq. (73)].

4) Enstrophy

Enstrophy is defined as
i1520-0469-55-22-3363-e28
with
i1520-0469-55-22-3363-e29
The previous developments now lead to the inner-product matrix SD:
i1520-0469-55-22-3363-e30

As in the case of the kinetic energy inner product, the enstrophy inner product is equivalent to the reference product 〈 · , · 〉0 in this problem. Basically, these equivalences come from two aspects: the 2D geometry makes the Y-derivatives disappear in the energy integrals and we keep looking at the geostrophic quantities, especially υg, so that we have simple relations between the norms and the perturbation geopotential ϕ. The latter approach is consistent with the quasigeostrophic assumption.

Finally, there are two sets of singular modes that are of interest at this stage: those linked with 〈 · , · 〉0 and those associated with 〈 · , · 〉A. The former are both maximizing kinetic energy and enstrophy, while the latter maximize the potential energy.

c. Linear and resolvent systems

1) The linear system

With the previously defined notations, equation system (9) can be written in matrix form:
i1520-0469-55-22-3363-e31
Since ϕEk, the horizontal derivative is equal to hϕ:
i1520-0469-55-22-3363-e32
The coefficients of the matrix may be developed:
i1520-0469-55-22-3363-e33
and we define the following nondimensional wavenumber:
i1520-0469-55-22-3363-e34
so that
i1520-0469-55-22-3363-e35
An equivalent expression can be found in Joly and Thorpe (1991) (see their Eq. 20). Operator L is the well-known linear system that governs the time evolution of any small perturbation in the Eady problem. The eigenvectors of L define the normal modes of the problem. Their determination can be found in Gill (1982), for example. Two cases are to be distinguished, for which the eigenvalues are either real or imaginary. They correspond to longwaves and shortwaves, respectively. The transition takes place at ω = ωc such that
ωcωc
The solutions read: shortwaves (ω > ωc): eigenvalues:
i1520-0469-55-22-3363-e37
 eigenvectors:
i1520-0469-55-22-3363-e38
longwaves (ω < ωc): eigenvalues:
i1520-0469-55-22-3363-e39
 eigenvectors:
i1520-0469-55-22-3363-e40

The eigenvectors are expressed in the basis (αk, βk). For shortwaves, the eigenvalues are imaginary, and the normal modes simply propagate at the top and bottom lids. In any couple of normal modes, one mode propagates westward while the other one propagates eastward, with no overall growth or decay in the modes’ amplitude. Nevertheless, the two modes (ϕ±ω) are not orthogonal with respect to any particular inner product, so that they may interact and temporarily achieve a significant amplification (Held 1985). Therefore, although the shortwave normal modes are neutral in time, a localized, temporary amplification is expected. In this context, the concept of singular modes gives the exact solution for obtaining the maximum amplification within our linear system. Of course, the same possibility exists for the longwave normal modes. In this case, one normal mode is growing exponentially while the other one is decaying.

2) The resolvent system

The next step is to determine the resolvent operator. Starting with relation (35), we can write
i1520-0469-55-22-3363-e41

For convenience, we note the resolvent M(0, T) = M. In a numerical model, this relation is solved by integrating along a predefined trajectory (which is stationary or not) the tangent linear version of the complete model code (which is nonlinear). In the Eady problem, however, the relation can be exactly known by the following technique (Joly and Thorpe, 1991): 1) diagonalize the matrix operator L, 2) integrate the diagonal system in time, and 3) express the resulting matrix in the (αk, βk) basis.

In particular, if we note P, the change of basis matrix from (αk, βk) to the normal mode basis (ϕ+ω, ϕω), then the following relation holds:
i1520-0469-55-22-3363-e42a
The diagonal matrix D is the resolvent matrix expressed in the normal-mode basis. Any linear initial mode can be expressed in the normal-mode basis:
ϕ0ckϕ+ωdkϕω
At time T, it will be mapped into
ϕTckeσTϕ+ωdkeσTϕω
This expression shows that every linear mode is integrated in time depending on the way it projects on the normal modes. Its amplification depends both on the growth rate σ and the time interval T. Also, one sees that M reduces to the identity matrix if T = 0 (no integration at all). At longwaves, we immediatly see that the growing normal mode ϕ+ω is amplified by a factor eσT over T, which shows simply that the resolvent matrix contains the information on the well-known e-folding times for unstable Eady modes. After calculations, the expression of the resolvent operator is obtained as a 2 × 2 matrix in the (αk, βk) basis. For longwaves, where ω < ωc,
i1520-0469-55-22-3363-e45
with
i1520-0469-55-22-3363-e46
Expression (45) is equivalent to Eq. (22) in Joly and Thorpe (1991). For shortwaves, where ω > ωc,
i1520-0469-55-22-3363-e48
with
i1520-0469-55-22-3363-e49

Again, for T = 0, M reduces to the identity matrix as expected by definition. As shown in section 2b, there are two interesting inner products to be tested. Moreover, the longwave and shortwave ranges have to be treated separatly. Thus, there are four different studies to be performed. In the next two paragraphs, the matrices of M are defined for each inner product.

3) Reference inner product 〈 · , · 〉0

Longwaves ω < ωc: We introduce (χk, ψk) by renormalizing [see section 2b, Eq. (18)] so that the matrix representation of M in the basis (χk, ψk) reads
i1520-0469-55-22-3363-e51

The singular modes are then obtained as the eigenvectors of the matrix M*M following the relations of the appendix.

At this point, the relations themselves become of little interest because of their complexity. Therefore, graphical outputs are preferred. This is the most suitable way for understanding the properties of the singular modes. We will discuss these figures in section 2d.

Shortwaves ω > ωc: From relation (48), we redefine the matrix representation of M in the (χk, ψk) basis:
i1520-0469-55-22-3363-e52

Note that the definitions of σ and γ are changed in the shortwave range compared with longwaves (see section 2c).

4) Potential energy inner product 〈 · , · 〉A

To determine MA, the matrix of M for the 〈 · , · 〉A inner product, we apply relation (A8) from the appendix. Therefore, we define the 2 × 2 matrix CA
i1520-0469-55-22-3363-e53
where C−1A is the square root matrix of SA [see Eq. (23)]. Then, MA reads
MAC−1AMCA

The singular modes are obtained as the eigenvectors of the matrix (MA)*MA, via the relations in the appendix.

d. Linear amplifications and singular modes

In this section, quantitative values are shown in order to give an insight into the analytical relations. The control parameters are chosen as indicated in Table 1.

1) Optimal linear amplifications μ1

Figures 1 and 2 show the amplifications for the reference inner product and the potential energy inner product, respectively. The values are plotted for several integration times T and against the dimensional wavenumber κh. In the following, we call NM the normal mode, SM1 the amplifying singular mode, and SM2 the damped singular mode. Therefore, the amplification of NM is exp(2σT), for SM1 it is μ1, for SM2 it is μ2, where μ1 and μ2 are the two eigenvalues of M*M.

In Fig. 1, we see that the overall aspect of the curves for μ1 is bell shaped: maximum optimal amplification is reached at a finite wavenumber, just as in the case of NM. At high wavenumbers (ωωc), all three amplifications tend toward 1. The Rossby height HR, which measures the vertical penetration depth of the modes, decreases so that there is no interaction between the surface and the top perturbations. Mathematically, the problem becomes equivalent to the Eady problem without a solid lid at the top, a case that supports no baroclinic instability. Thus, the singular modes are just as inefficient as the normal modes.

At “medium wavenumbers” ωc/2 ⩽ ω ⩽ 2ωn, three points are to be stressed:

  • A maximum for μ1 is reached at a finite wavenumber ωs with ωs > ωn. For T = 6h, ωs corresponds to a singular mode of wavelength 3000 km. The normal mode, instead, has a 4000-km wavelength. For larger values of T, ωs decreases slowly. For very long T, ωs becomes close to the optimal NM wavenumber ωn.
  • In the vicinity of ωc, NM becomes a neutral mode, whereas SM1 is still clearly amplifying. Thus, SM1 is 4 to 5 times more amplified than NM at T = 24 h, 10 to 15 times at T = 48 h and about 20 times at T = 72 h. One can obtain strongly amplifying linear modes where the classical normal mode approach predicts no amplification at all. Note that SM1 depends on T, so that we do not necessarily have similar singular mode solutions for different intervals T (see next paragraph).
  • For big intervals of T, ωs tends toward the NM wavenumber ωn but μ1 does not tend toward the NM amplification. There is a difference of about 20% between μ1 and exp(2σT), up to T = 360 h (not shown). This difference is due to the fact that the normal mode is never lying exactly in the direction of the SM1 mode, for whatever T is considered. This result underlines the fact that the problem of finding the singular modes with respect to a given inner product is not equivalent to the problem of finding a normal mode that would amplify exponentially in time even at “infinite” timescales.

Thus, the interaction between two neutral waves (ω > ωc) can be constructive for kinetic energy. This shows that the Eady problem sustains growing shortwaves over a meteorologically significative period. This point was already noticed by Rotunno and Fantini (1989). They choose two neutral three-dimensional waves so that their initial surface pressure perturbation is zero. Thus, their wave is suboptimal in the sense of kinetic energy, for example. Note that the solutions for three-dimensional perturbations can be compared qualitatively to the two-dimensional ones, since the search for singular vectors remains a two-dimensional problem. Also, these authors note that two neutral normal modes, which have wavenumbers close to the critical ωc, can produce growth almost in the same way as unstable normal modes over a given, long period. This result is extended here to singular modes since the μ1 (and μ2) curve is perfectly smooth when ω passes through ωc. This property is reminiscent of the works of O’Brien (1992) and Davies and Bishop (1994), which both show curves of optimal instantaneous growth rates that have this smooth bell-shaped pattern. Moreover, the structure of SM1 is similar on both sides of ωc (not shown): there is no singularity such as a cutoff behavior appearing at medium wavenumbers for SM1 and SM2.

At small wavenumbers (ωωc), the μ1-curve becomes close to the NM curve: the normal mode has nearly the optimal amplification. Furthermore, SM1 and NM (and SM2) become more and more neutral as ω decreases.

Figure 2 shows the amplifications with respect to the potential energy inner product 〈 · , · 〉A. At high wavenumbers, all the amplifications are about 1. Again, the Rossby height is too small and no storage of perturbation potential energy A is possible. Short SM1 waves behave the same with respect to 〈 · , · 〉0 and 〈 · , · 〉A: they are neutral.

At medium wavenumbers, the μ1-curve is concave and μ1 is only slightly bigger than the NM amplification. Thus, SM1 is not much more efficient than NM in generating potential energy A. Inversely, one may say that NM is nearly optimally generating A.

At small wavenumbers, the μ1-curve is increasing dramatically and values for μ1 become unbounded: the longer the wave, the stronger the amplification of A. This behavior is contrary to the one detected for kinetic energy. Thus, long SM1 modes are efficient at extracting potential energy from the mean flow, but they are inefficient at converting this energy into kinetic energy. This point is further discussed in the next paragraph.

2) Structures of the singular modes

Figures 3, 4, 5, and 6 show the structure of the geostrophic meridional wind υg for several types of modes. They all deal with normal modes and singular modes. An overall examination shows that all those modes have a westward-tilted wind field. For quasigeostrophic, baroclinic perturbations, this is the crucial property for converting mean state potential energy  into perturbation potential energy A, and then into perturbation kinetic energy K (e.g., Holton 1979). Furthermore, we recall that a northward heat flux is needed to convert  into A, and a positive vertical heat flux is necessary to convert A into K. These two conversions are the only ones that are at work in the 2D Eady problem. Thus, the tilted structures are all consistent with modes that grow from their initial state. However, unlike normal modes, singular modes do not amplify at a constant rate in time. In fact, they are always shaped in such a way as to maximize the corresponding energy production (A or K) over the time interval T considered. Furthermore, all SM1 modes exhibit a symmetric structure with an equally strong anomaly at the surface and at the top boundaries at initial time. This pattern is to be compared with the results of Mukougawa and Ikeda (1994). In their cases with nonuniform potential vorticity, the singular modes have maximum amplitudes inside the domain, with a more or less pronounced foliated structure. In the present study, the constraint of uniform interior potential vorticity forces maximum gradients to appear at the two boundaries.

Figure 3 shows a comparison between the NM structures and the SM1 structures for 〈 · , · 〉0 and T = 6 h. Unlike the NM structure, which becomes more and more vertical as the wavelength decreases, the SM1 structure always shows a clear westward tilt. The phase shift between the surface and the top anomaly is about L/4 at all wavelengths. This shift allows both for a positive domain-integrated northward heat flux [υgθ] and a positive vertical heat flux []. Therefore, the SM1 modes are efficient both at ÂA and AK conversions. Note that, for longwaves, the SM1 structure is similar to the NM structure.

Figure 4 shows the SM1 for 〈 · , · 〉0 and T = 48 h. There is no visible difference in the structures for longwaves when compared to the T = 6 h solutions. This is consistent with the fact that the SM1 is close to the NM mode at long wavelengths. Thus, because the NM mode does not change its spatial structure with time, the SM1 mode also keeps a similar shape. For medium and short waves, the westward tilt is stronger for the T = 48 h solution than for the T = 6 h solution. This behavior is a compensation for the clockwise rotation due to the differential advective effect of the mean flow onto the wave. As a consequence, the SM1 departs significantly from the NM mode, so that the latter always has a 20% loss in amplification when compared to the former. This kind of “reshaping” with T illustrates physically why the SM1 and NM mode never match exactly at medium waves for any T (see previous paragraph).

Figure 5 shows the SM1 structure for 〈 · , · 〉A and T = 6 h. For medium and short wavelengths, the structure is quite similar to the SM1 structure for the reference inner product. At longwaves, the westward tilt is smaller than for the 〈 · , · 〉0-SM1.

Figure 6 shows the SM1 structure for 〈 · , · 〉A and T = 48 h. At shortwaves, the structure of the SM1 is similar to the structure for the 〈 · , · 〉0 product, whereas at medium waves, the SM1 mode is rather similar to the NM mode. In the latter case, the westward tilt is smaller than the tilt for the 〈 · , · 〉0-SM1. At longwaves, the 〈 · , · 〉A-SM1 has almost a vertical structure so that the SM1 maximizes the positive correlation between υg and θ in order to get a positive northward heat flux. This correlation is achieved by creating singular modes with the top anomaly almost overhead of the surface anomaly. At the same time, this configuration inhibits baroclinic vertical fluxes because the shape of the mode remains rather “barotropic.” Thus, the AK conversion is poor and perturbation energy is simply stored in the A reservoir. This result is consistent with the amplifications presented previously: the longwave 〈 · , · 〉A-SM1s are very efficient (their amplification μ1 is “explosive”) whereas the 〈 · , · 〉0-SM1s are not (the corresponding μ1 decreases with ω).

One should ask why 〈 · , · 〉A singular modes, at longwaves, can amplify nearly infinitely? First, the transverse, meridional front ∂θ/∂Y has an infinite length in Y. Therefore, the basic-state potential energy reservoir  is infinitely large. Second, a closer inspection of the resolvent MA for ω → 0 shows that the amplification μ1 tends to m(T)2/ω2 with m(T) = 3fΛT/N. While the perturbation potential energy of the SM1-mode is forced to be 〈x1, x1A = 1, its kinetic energy scales like 〈x1, x1K ∝ 1/ω2, and very large meridional winds are obtained (υg ∝ 1/ω). This behavior is noted by Bretherton (1966) for a two-layer model. To be specific, a normalized potential temperature wave can sustain very large heat fluxes at very small wavenumbers, and can therefore extract large amounts of energy from the basic-state reservoir Â, which is itself infinite. Davies and Bishop (1994) have observed a similar behavior on their optimal instantaneous growth rate for potential temperature: in the pure 2D case, the growth rate boosts to infinite values, while it remains bounded when a finite cross-front wavelength is prescribed (see their Fig. 3). Also, in Joly (1995), the amplification curve for the 〈 · , · 〉A product has the shape of a low-pass filter (see his Fig. 2). Nevertheless, the 〈 · , · 〉A amplifications are not increasing for small wavenumbers as in our analytical case. This difference comes from the Y-dependence introduced in Joly’s front: the  reservoir is limited and very long waves cannot develop over significant timescales. Note that we only compare here those singular modes that are found baroclinic in Joly’s numerical study.

3. Error growth

a. Definitions

In this section, some considerations concerning the time evolution of error covariances are addressed. This dynamical study provides nonobvious error structures that are fitted to the flow. These types of error structure are nowadays being implemented in most data assimilation systems because they provide a more realistic error field than conventional isotropic structures. Note that singular vectors provide information on the most rapidly growing error features (see section 3b), so that they are useful in sensitivity studies. We first recall briefly some standard results as they are used in the Kalman filter theory (Kalman 1960; Kalman and Bucy 1961; Ghil 1989). The model variable of our problem is ϕ, expressed in the spectral space Ek as defined in section 2a and 2b. We consider an error ϕ′ attached to this field. If E[ · ] stands for the statistical mean, then we have
i1520-0469-55-22-3363-e55
where Pt stands for the error covariance matrix in the spectral space Ek. Thus, Pt gives the second-order error statistics on the two coefficients ak and bk, in the functional basis (χk, ψk). The time evolution of the error is the sum of the dynamical growth of the initial error and the model error:
ϕMϕ0ϕ
with ϕ" the model error. The time integration of the error covariances then reads
Pt = MP0M* + Q and Q = E[ϕ"ϕ"t]
with Q the model error covariance matrix. This is the standard equation for advancing in time the error covariances in the Kalman filter. Three remarks are to be made at this point:
  • the adjoint of M is with respect to the reference inner product 〈 · , · 〉0, since the basis (χk, ψk) has been chosen;
  • Q = 0, that is, the perfect model assumption is retained;
  • an important conceptual difference is the role played by the time variable t. It is now actually a variable, whereas in the previous chapter, we introduced a (fixed) parameter T that was only control.

b. Spectral coefficients

In this section, basic results are derived from the developments of the previous chapter. This discussion is devoted to the general behavior of error covariances and it is not intended to introduce a complete Kalman filter. Therefore, the initial error covariance matrix P0 is set to the identity matrix I. This means that errors on the two spectral coefficients are supposed to be uncorrelated, and that both variances are equal. This assumption allows for a maximum overlap with the equations already calculated in the previous chapter.

The covariance evolution now reads
PtMM

The rhs operator has the same eigenvalues as M*M:μ1 and μ2. This property underlines the role played by μ1 and μ2 in the context of linear error growth: they are the Lyapunov coefficients along the prescribed trajectory. If a specific P0-matrix is defined, then this result is altered and the largest and smallest eigenvalues of P0 must be taken into account.

A closer inspection of μ1 and μ2 shows that
ω,t,μ1μ2Pt
for the 〈 · , · 〉0 amplifications. Thus, for a fixed wavenumber, an approximate one-dimensional equation for the covariances is obtained by only considering the dominating eigenvalue and its eigenvector.
The approximate matrix reduces to the scalar μ1 and explains a fraction of the total variance:
i1520-0469-55-22-3363-e61

If the whole spectrum of wavenumbers is considered, then a subset of growing eigenvectors of Pt is used in the approximation. From Fig. 1 and for small t, it is seen that the explained variance will be rather poor due to the flattening of the spectrum of μ1. Thus, a large number of eigenvectors is required to explain a significant amount of variance for small lead time (e.g., t = 6 h).

This technique is now commonly proposed for simplifying the covariance dynamics in a numerical model Kalman filter because it allows for decreasing the number of dimensions of the system and thus saves computer time and memory (see Bouttier 1994). In the Eady problem, it would be efficient at medium wavenumbers where μ1 significantly departs from 1 (see Fig. 1).

Houtekamer (1995) studies the linear modes that maximize the forecast errors in a quasigeostrophic, three-layer model. In this study, the first n0 eigenvectors from the forecast error covariance matrix are extracted. When only the first eigenvector is retained (n0 = 1) for a prescribed integration time t = T, then the explained variance is indeed maximum at forecast time T. Houtekamer suggests that the longer the T, the fewer eigenvectors are needed to explain a given fraction of variance. This behavior is similar to the Eady problem at medium wavenumbers since the spectrum of μ1 becomes more and more bell shaped with increasing T.

c. Physical space

This section is dedicated to an overall study of error variances in physical space, that is, in the vertical X–Z plane of the Eady problem. We now concentrate the study on the quasigeostrophic geopotential φ. Absolute values are given in m2 s−2.

The physical geopotential φ at a point M is linked to the spectral geopotential ϕ by the relation
i1520-0469-55-22-3363-e62
where ϕEk is any solution to the Eady problem (see section 2a). We suppose that the error statistics on ϕ are given by the matrix Pt introduced in section 3b for any time t. In particular, the simplified relation Pt = MM* is kept.
Let Π be the covariance matrix of φ, then
ΠEφMφtM
where M and M′ are two arbitrary points.

Equation (63) gives the covariances of the geopotential in physical space. It can be interpreted either as a quadratic function of M (if M = M′, then the physical variances are obtained), or as a linear function of M′ (if M is fixed, then the physical covariances of all points M′ with M are obtained).

1) Variances in the shortwave range

If M = M′, then relation (63) leads to the expression for the physical space variances. As an illustration, the following expression gives the variances at midtroposphere (Z = 0):
i1520-0469-55-22-3363-e64

Thus, variances can be split into three contributions:

  • a steady, periodic structure where the spatial wavelength is L/2, half the channel length;
  • a pulsating, nonpropagating term that corresponds to a standing wave. (its time period is half the period of the normal mode);
  • two propagating waves that have the same phase speeds as the normal modes, with opposite signs.

Figures 7a,b show the variances for L = 2222 km, at 0 and 24 h, respectively. First, note that the initial conditions on the spectral covariances P0 = I lead to an initial structure in physical space with a vertical axis of zero variance located at X = L/4, which is repeated at X = 3L/4 because of the quadratic nature of Π. This behavior may crudely be considered as the result of the assimilation of perfect data at Z = ±H and X = L/4 in a previous analysis step. An axis of maximum variance is initially located at X = 0 and X = L/2. For a given X, minimum initial variance is always located at midtroposphere. With time, the impact of the two propagating terms can be seen, as the kernels of maximum variance propagate eastward at the surface and westward at the tropopause. Thus, the initial variance field is restored at about 72 h (not shown). The standing wave term produces an overall pulsation with a period of about 72 h. Therefore, the combined effects of the pulsation and the propagation lead to an absolute maximum variance of about 10.8 at 36 h and at X = L/4 and X = 3L/4. This value is to be compared with the maximum initial variance of 6.0. Also, at 36 h a vertical axis of zero variance appears at X = 0 and X = L/2. Thus, if we note τ = 2π/σ, the advection time of the neutral normal modes in the channel, then the period for the variances is τ/2 and the configuration with maximum absolute variance is obtained at τ/4. The variances rise significantly over the latter interval, although the trajectory supports no amplifying normal modes. This behavior is consistent, however, with the existence of other amplified linear modes.

2) Variances in the longwave range

At longwaves and at midtroposphere, the variances read
i1520-0469-55-22-3363-e65

Is is evident from (65) that there is no propagation at longwaves. Nevertheless, a spatial structure, F(X, Z), is emerging while σt grows and a shift takes place from the initial structure to this long-term limit. This shift is the longwave counterpart to the propagating and the standing wave evolutions for the shortwaves. With asymptotic studies, t → +∞ and ω → 0, it is seen that the “final” structure F(X, Z) tends to a parabolic function with Z and always bears a “wavy” aspect (the minimum of variance is located alternatively above and below the Z = 0 level).

Figures 7c,d show the variances for L = 4000 km, at 0 and 24 h, respectively. In contrast to the shortwave case, there is no oscillatory behavior: the variances grow steadily. The maximum variance is, however, not growing very quickly over the first 24 h, which is consistent with the rather long e-folding time for the normal mode (1/σ ≈ 5 days at L = 4000 km). The general pattern is tending to a wavy structure with maximum variance at both boundaries and minimum error variance at midtroposphere.

4. Conclusions

In this paper, the concept of singular modes is revisited in the two-dimensional Eady problem with uniform potential vorticity. A complete analytical study is performed for different norms that have a physical meaning: potential energy, kinetic energy, and enstrophy. However, the latter two are equivalent in our formulation because of the simplified, 2D geometry. An overview of the results concerning the singular modes finally shows the following.

  1. In contrast to the behavior of normal modes, singular modes do not exhibit any cutoff: their amplifications and shapes vary continuously over all wavenumbers. One consequence is that significant amplifications are obtained in the range of neutral normal modes for both kinetic and potential energy. This result is not new: previous studies underlined the existence of growing linear modes at shortwaves (see Rotunno and Fantini 1989). Our study, however, gives the exact bounds for these amplifications.
  2. For the kinetic energy norm, a slight scale selection is observed: for all wavenumbers, there is one finite wavenumber (ωs) for which the corresponding singular mode has the greatest amplification. It should be stressed, however, that the peak in ωs is not as sharp as the one for ωn (a point that was noticed by Joly 1995). Thus, a wide range of horizontal wavelength is possible, which indicates that this framework is barely able to predict a favored horizontal scale for a singular mode. With increased time intervals T, the singular mode “optimal” wavenumber ωs tends to the optimal normal mode wavenumber (ωn), though the structure of the singular mode itself does not exactly tend to the structure of the normal mode even if large time intervals are considered. All singular modes present a clear westward tilt, so that energy conversion from the basic-state potential energy to the perturbation kinetic energy reservoir is favored.
  3. For the potential energy norm, no scale selection is observed: the longer the singular wave, the higher its amplification. This lack of scale selection is due to the strong coupling of the two boundaries at large wavelengths and to the infinite amount of potential energy in the basic state. Furthermore, for long time intervals T, the singular modes bear a quasi-vertical shape, with top and bottom anomalies overhead.
  4. The potential energy and kinetic energy singular modes only look the same when large wavenumbers (ω greater than ωs or ωc) are considered.

Although a variety of singular modes are presented in this paper, one crucial question remains unanswered:what are the relevant singular vectors in a realistic atmosphere? This question goes beyond the scope of this work, but some results may provide guidelines for its answer. First, recall the basic definition of a singular vector (see section 1 or the appendix) where it is evident that a solution to the maximization problem always exists for any norm with a physically acceptable shape (provided the norm bears some physical significance). Furthermore, the results show that optimal growth is much more scale-insensitive than exponential growth, for any norm. Therefore, it is felt that the norm itself is not the relevant point, and only the initial and large-scale conditions are important. Thus, the use of singular vectors would consist, in a realistic case, to locate preexisting anomalies (surface potential temperature, tropopause foldings) and to estimate basic parameters of the baroclinic flow: shear Λ, stability N2, and Rossby height HR. Then, one could determine which kind of quadratic quantity can be amplified by the total perturbation. Also, an objective measure of any amplification is provided by computing the ratio of the actual growth to the dominating singular value μ1. For the study of singular vectors with two interacting boundary Rossby waves, the results in section 2d suggest that the 〈 · , · 〉0 product is best suited since it takes into account the internal baroclinic conversion between perturbation potential and kinetic energy.

In this study, we take advantage of the similarity between the singular mode and the Kalman filter calculations to perform a particular study of error variance dynamics. We set the initial covariance matrix to identity so that the eigenvalues μ1 and μ2 become the bounds for the norm of the covariance matrix in spectral space. An exact equation is immediately derived for measuring the loss of explained variance if only the first eigenvector is retained in the covariance matrix. It is then seen that the approximation is actually better when long time integrations are considered. For short time intervals, one needs a large set of wavenumbers to explain the error variance, due to the flattening of the amplification spectrum.

Error variances in physical space are also presented. It is shown that variances might rise significantly even in the range of neutral normal modes because of conjugate pulsating and propagating effects. For long or short waves, the typical timescales for variances are closely related to a fraction of the e-folding time or the period of neutral modes, respectively. From a more general point of view, the knowledge of the normal modes is still important in order to determine the overall behavior of error variances (e-folding times, periods, and phase speeds).

As far as data assimilation is concerned, current work concerns the Observing System Simulation Experiments (OSSEs) (see Snyder 1996). For example, in idealized OSSEs, data assimilation systems are simulated in quasi-geostrophic or semigeostrophic models. Usually, these models have uniform interior potential vorticity, which makes them work in simple, spectral spaces (such as in our Eady problem) and with relatively small state vectors. These experiments typically contain a step in which singular modes are searched. In this numerical framework, the possibility of defining physically different singular vectors, by using different norms, can be advantageous. Indeed, while the development of synoptic-scale cyclones is well understood, much remains unknown on the incipient stages of mesoscale cyclones. This means that one can test several initial structures that fulfill different criteria for growth in an idealized flow. Furthermore, since the balanced models have small state vectors, error covariances can be predicted with a Kalman filter. Thus, this work is hoped to provide a simple guideline for such kind of numerical results where there is a greater complexity (time-dependent basic states and scale separations).

Acknowledgments

This work benefited from useful comments by Pierre Bernardet and Dave Parsons. Alain Joly and François Lalaurette were involved in general discussions on the topic. The anonymous reviewers also provided constructive comments. The author is thankful to the Ecole Nationale de la Météorologie, Météo-France, which allowed for a longer stay at the Centre National de la Météorologie.

REFERENCES

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  • Davies, H., and C. Bishop, 1994: Eady edge waves and rapid development. J. Atmos. Sci.,51, 1930–1946.

  • Eady, E., 1949: Long waves and cyclone waves. Tellus,1, 33–52.

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  • ——, 1984: Modal and non-modal baroclinic waves. J. Atmos. Sci.,41, 668–673.

  • ——, 1988: Optimal excitation of neutral Rossby waves. J. Atmos. Sci.,45, 163–172.

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  • Gill, A., 1982: Atmosphere–Ocean Dynamics. International Geophysics Series, Vol. 30, Academic Press, 662 pp.

  • Held, I. M., 1985: Pseudomomentum and the orthogonality of modes in shear flows. J. Atmos. Sci.,42, 2280–2288.

  • Holton, J. R., 1979: An Introduction to Dynamic Meteorology. International Geophysics Series, Vol. 23, Academic Press, 391 pp.

  • Houtekamer, P., 1995: The construction of optimal perturbations. Mon. Wea. Rev.,123, 2888–2898.

  • Joly, A., 1995: The stability of steady fronts and the adjoint method:Nonmodal frontal waves. J. Atmos. Sci.,52, 3082–3108.

  • ——, and A. J. Thorpe, 1991: The stability of time-dependent flows:An application to fronts in developing baroclinic waves. J. Atmos. Sci.,48, 163–182.

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APPENDIX

Expression of the Singular Modes of a 2D Linear Transformation

Let E be the vector space of dimension 2 that contains all the couples of complex numbers in the form x = (cx, dx) with cx and dx complex. We write 〈x, y〉 the canonical inner product between two vectors x and y in E:
i1520-0469-55-22-3363-ea1
We now consider a 2 × 2 complex matrix F, noted
i1520-0469-55-22-3363-ea2
The adjoint matrix F* is readily obtained by taking the Hermitian conjugate of F:
i1520-0469-55-22-3363-ea3
According to the definition of singular vectors, as introduced in section 1, the singular modes associated to the matrix F are the eigenvectors of F*F. The problem now reduces to solving the diagonalization of the product matrix F*F, where F is given by (A2). Thus, we solve the eigenvalue problem det(F*FμI) = 0. The solutions for the eigenvalues μ1 and μ2, as well as for the eigenvectors (x1, y2), finally read
i1520-0469-55-22-3363-ea4
and the following notations have been introduced:
i1520-0469-55-22-3363-ea6
The above developments are valid if the canonical inner product is considered. They can be generalized to any inner product, represented by a positive-definite 2 × 2 matrix S such as
i1520-0469-55-22-3363-ea7
Formula (A2) must then be replaced by
FSC−1SFCSC−1S2S
where C−1S is the 2 × 2 square root matrix of S.

As a final conclusion, it is emphasized that the previous developments are tractable in a two-dimensional framework. However, as soon as a three-dimensional problem is considered, these developments become unrealistically huge. Therefore, these results must be regarded as an easy way of solving an idealized two-dimensional problem, but they are unsuitable for more complex problems for which numerical algorithms are needed.

Fig. 1.
Fig. 1.

Linear amplifications for the reference inner product 〈 · , · 〉0 versus wavenumber: normal mode (solid), optimal singular mode μ1 (dashed), and damped singular mode μ2 (dashed–dotted). Different timescales are shown: (a) T = 6 h, (b) T = 24 h, (c) T = 48 h, (d) T = 72 h. In (a), arrows indicate the location of particular wavenumbers: most amplified normal mode ωn, most amplified singular mode ωs, and smallest neutral normal mode ωc.

Citation: Journal of the Atmospheric Sciences 55, 22; 10.1175/1520-0469(1998)055<3363:LAAEGI>2.0.CO;2

Fig. 2.
Fig. 2.

Linear amplifications for the potential energy inner product 〈 · , · 〉A versus wavenumber: normal mode (solid), optimal singular mode μ1 (dashed), and damped singular mode μ2 (dashed–dotted). Different timescales are shown: (a) T = 6 h and (b) T = 24 h.

Citation: Journal of the Atmospheric Sciences 55, 22; 10.1175/1520-0469(1998)055<3363:LAAEGI>2.0.CO;2

Fig. 3.
Fig. 3.

Structure of the SM1 and NM geostrophic meridional wind υg: normal mode NM (left) and singular mode SM1 for the reference inner product 〈 · , · 〉0 (right). The timescale is T = 6 h. Note also that the neutral NM is the one with maximum intensity at the surface.

Citation: Journal of the Atmospheric Sciences 55, 22; 10.1175/1520-0469(1998)055<3363:LAAEGI>2.0.CO;2

Fig. 4.
Fig. 4.

Structure of the SM1 geostrophic meridional wind υg: the singular mode SM1 is for the reference inner product 〈 · , · 〉0. The timescale is T = 48 h.

Citation: Journal of the Atmospheric Sciences 55, 22; 10.1175/1520-0469(1998)055<3363:LAAEGI>2.0.CO;2

Fig. 5.
Fig. 5.

Structure of the SM1 geostrophic meridional wind υg: the singular mode SM1 is for the potential energy inner product 〈 · , · 〉A. The timescale is T = 6 h.

Citation: Journal of the Atmospheric Sciences 55, 22; 10.1175/1520-0469(1998)055<3363:LAAEGI>2.0.CO;2

Fig. 6.
Fig. 6.

Structure of the SM1 geostrophic meridional wind υg: the singular mode SM1 is for the potential energy inner product 〈 · , · 〉A. The timescale is T = 48 h.

Citation: Journal of the Atmospheric Sciences 55, 22; 10.1175/1520-0469(1998)055<3363:LAAEGI>2.0.CO;2

Fig. 7.
Fig. 7.

(a), (c) Variances of geopotential in physical space at initial time, and (b), (d) at day 1 for a (a), (b) shortwave and a (c), (d) longwave. Values are in m4 s−4. Maximum values are also indicated (arrows).

Citation: Journal of the Atmospheric Sciences 55, 22; 10.1175/1520-0469(1998)055<3363:LAAEGI>2.0.CO;2

Table 1.

Values of the physical constants for the quantitative application.

Table 1.
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