On the definition of Landau constants in amplitude equations away from a critical point

A weakly nonlinear stability analysis of shear flows based on amplitude expansion is re-examined. While it has been known that the condition required to define the coefficients of the resulting Stuart–Landau series representing the nonlinear temporal evolution of the most amplified Fourier component of a disturbance is not unique, we show that it can be formulated in a flexible generic form that incorporates different conditions used by various authors previously. The new formulation is interpreted from the point of view of low-dimensional projection of a full solution of a problem onto the space spanned by the basic flow vector and the eigenvector of the linearized problem. It is rigorously proven that the generalized condition formulated in this work reduces to a standard solvability condition at the critical point, where the basic flow first becomes unstable with respect to infinitesimal disturbances, and that it results in a well-posed problem for the determination of coefficients of Stuart–Landau series both at the critical point and a finite distance away from it. On a practical side, the generalized condition reported here enables one to choose the projection in such a way that the resulting low-dimensional approximate solution emphasizes specific physical features of interest via selecting the appropriate projection weight matrix without changing the overall asymptotic expansion procedure.


Introduction
Weakly nonlinear stability analysis of fluid flows is a mature area of research. It was introduced in its most well-known form in the 1960s in pioneering works [1,2] with a number of notable contributions throughout the 1970s, see [3][4][5][6]  basic flow, which is a relatively simple solution of the governing equations obtained analytically or numerically, and perturbations. Linearization of a problem about the basic flow results in an eigenvalue problem for perturbations. Upon a suitable discretization a problem containing differential operators is typically converted to a generalized algebraic eigenvalue problem and the resulting eigenvectors determine the spatial distribution of perturbed flow quantities. However, the magnitude of such disturbances cannot be determined within the framework of a linearized problem so that the perturbation amplitude is taken as an unknown time-dependent factor multiplying the spatial eigenfunction. One of the main goals of a weakly nonlinear analysis is to model the temporal evolution of the disturbance amplitude A(t) and to determine whether and at what value it saturates. This leads to the derivation of the so-called amplitude equations of Stuart-Landau type [1,7] dA dt ¼ sA þ A X 1 k¼1 K k jAj 2k : (1:1) There have been suggested several ways for such a derivation with one of the most common methods using the multiple time scale expansion approach [8]. More recently, procedures based on a centre manifold reduction were also used and shown to produce identical results, however, at a much higher computational cost comparable with that of direct numerical computations, see [9]. This frequently makes the multiple scale expansion the method of choice when relatively inexpensive practical results in the finite neighbourhood of a critical point are of interest. A crucial point in developing weakly nonlinear analysis is the choice of a small parameter that is used for producing asymptotic series approximating the full nonlinear solution of the problem. The relative parametric distance kR 2 R c k/kR c k ( 1, where R and R c represent the sets of problems' governing parameters at which the solution is required and of their critical values at which the basic flow first becomes unstable with respect to infinitesimal disturbances, respectively, is frequently chosen as the expansion parameter following the pioneering work [1]. This enforces a specific scaling of solution quantities near the critical point. In particular, the amplification rate of infinitesimal disturbances needs to be proportional to kR 2 R c k/kR c k. The validity of such a rigid scaling is expected in the asymptotic vicinity of the critical point but cannot be guaranteed in practically important regimes characterized by small but finite values of kR 2 R c k/kR c k. Moreover, such a parameter cannot be introduced in principle for flows that are always linearly stable such as pipe Poiseuille or plane Couette flows [10]. From a physical point of view, it appears more natural to take the magnitude of the perturbation amplitude itself as a small parameter. This leads to the so-called amplitude expansion [2,10]. It has been shown that its introduction does not rely on the proximity of the critical point or any rigid a priori scaling assumptions on solution quantities. In particular, in this approach the amplification rate of infinitesimal disturbances may happen to be, but does not have to be asymptotically small. The only validity condition for such a procedure is imposed by the requirement that the perturbation amplitude remains sufficiently small so that its powers form a set of elements with magnitudes progressively decreasing at a sufficiently fast rate to ensure the meaningful truncation of the asymptotic series (1.1) [11][12][13].
Despite this apparent advantage of using amplitude expansion there exists a very large body of literature, where parametric expansions about the critical point are favoured as the main tool of weakly nonlinear analysis. A comprehensive review of this field is outside the scope of the present paper, but we point an interested reader to studies reported in [8,[14][15][16], where the comparison of parametric and amplitude expansion methods demonstrating the equivalence of various approaches near the critical point can be found and the methods are presented in a way similar to the amplitude expansion summarized in §2. One of the reasons the parametric expansion first introduced in [1] remains a popular choice is that algebraically it enables one to estimate the coefficients of the Stuart-Landau series (1.1) modelling a temporal evolution of the disturbance amplitude using the information from a critical point alone by employing the uniquely defined solvability condition (see equation (2.23) in §2.3). This, however, comes at a price of a reduced convergence range [17]. The parametric expansion approach about a critical point also suffers from a subtle inherent inefficiency. Before the method can be applied, the critical point in the problem parameter space has to be found. This is typically done iteratively by solving the linearized eigenvalue problem over a range of the parameter values R until R c is found. Therefore, the eigenfunctions are readily available at the parametric point of interest R and can be used as the basis of the amplitude expansion. However, in order to cast the problem in the form suitable for the application of the solvability condition this readily available eigenfunction of interest is ignored and replaced by the one computed at the critical point.
By contrast, the amplitude expansion procedure summarized in §2 allows using the eigenfunction computed for the parametric values of interest away from a critical point directly. This, however, introduces the ambiguity demonstrated in §2.3 in supercritical regimes the definition of the eigenfunction amplitude becomes non-unique and requires an additional condition to be fixed. Such an ambiguity has been recognized from the inception of the method [2] and various authors proposed different ways of fixing it [9][10][11]13,18]. In the present paper, we show that previous individual attempts of fixing this ambiguity can be cast in a flexible general form that has a straightforward interpretation in the context of projecting fully nonlinear solutions onto the space spanned by the vector of the basic flow and the eigenvector of the linearized problem. The procedure enables one to effectively choose 'the projection angle' in such a way that the resulting lowdimensional solution is optimized to emphasize the desired aspects of the full solution. From a practical point of view, this, perhaps, is the main feature of the procedure described in this paper. We start with the idea first formulated in [19] and subsequently prove that the weakly nonlinear stability problem complemented with the specific projection criterion is well-posed at the critical point and away from it. Therefore, Landau constants in the Stuart-Landau series (1.1) can be efficiently evaluated taking into account the chosen meaning of the amplitude.
2. Summary of the expansion formalism and properties of the resulting operators 2.1. Basic flow Consider a system of governing equations of Navier-Stokes type written in an appropriately nondimensionalized form. Such a system of equations includes linear and nonlinear (at least quadratic) terms and represents the fundamental physical principles of conservation of momentum, mass and thermal energy and can include constitutive equations describing various fluid properties and processes taking place in it. The actual form of equations depends on the physical problem at hand and is not of importance for the analysis presented in this paper, but we will refer to the following set of non-dimensional equations describing a two-dimensional fluid flow between two infinitely wide and long differentially heated vertical plates, see [12] and §6 for details, as an illustration of the overall equation structure that is taken into account in the subsequent derivations: and @T @t þ u Á rT ¼ 1 GrPr r 2 T: (2:2) Here u ¼ (u, v) is the vector of fluid velocity, p is the pressure, T is the fluid temperature, g ¼ (0, À1) is the unit vector in the direction of gravity and R ¼ (Gr, Pr) is the set of the governing non-dimensional parameters (Grashof and Prandtl numbers in this example). The fluid flow is considered in a domain that is bounded in at least one spatial (21 x 1 in the above example) direction and unbounded in at least one other (21 , y , 1 in the above example) direction. Assume that the above equations subjected to appropriate boundary conditions have a 'simple' steady solution w 00 ¼ (u 00 , T 00 , p 00 ) T referred to as the basic flow. By 'simple', it is typically meant that for any fixed set of the governing parameter values R such a solution depends on the spatial coordinate in the direction of finite system extent (x), but not the one extending to infinity (translational symmetry in y). Symbolically, this is written as w 00 ¼ w 00 (x; R).

Linear stability and operators
Once the basic flow is found the question arises whether it is stable with respect to infinitesimal perturbations. In spatially extended systems, they can be conveniently given in the normal form A(t)w 11 (x)E þ c:c:, where A(t) is the time-dependent complex amplitude of disturbances with time evolution generally given by the term E ; exp(iay) signifies the fact that the disturbance is 2p/a periodic in the extended direction, a is the spatial wavenumber and c.c. stands for complex conjugate. Substituting rsos.royalsocietypublishing.org R. Soc. open sci. 5: 180746 into (2.1) and neglecting terms containing higher powers of amplitude one obtains a system of linear partial differential equations that can be written in a matrix operator form as Here matrix operator A a;R represents terms involving spatial derivatives. This operator also contains governing parameters R appearing in the original equations. Matrix B arises from terms containing time derivatives so that for time-dependent problems Bw 11 = 0. Typically, the governing equations can be written in such a form that this matrix does not contain the governing parameters, but it is singular as some of the governing equations, such as the continuity equation in system (2.1) does not contain time derivative explicitly. Thus B contains zero rows. This, however, does not affect the proofs and discussions presented in the subsequent sections as long as the condition Bw 11 = 0 remains valid. In practice, equation (2.5) is solved numerically upon adopting a suitable discretization scheme for approximating spatial derivatives. As a result, matrix differential operators A a;R and B are converted to square N Â N matrices with complex elements, where size N depends on the number of individual scalar functions forming solution w and the number of discretization points for each of them. Thus in what follows we will treat these operators as regular matrices and solution vectors as regular vectors. Equation (2.5) can only be satisfied for any time if F(A) ¼ sA that is and where s is referred to as the complex amplification rate: s ¼ s R þ is I . Equation (2.5) is then rewritten in the operator form as (A a;R À sB)w 11 ; L a,s;R w 11 ¼ 0: The above has a non-trivial solution w 11 = 0 only if s is the eigenvalue of the operator L a,s;R . When the problem domain is bounded (e.g. in x in the illustrative example considered above) the eigenvalue spectrum is discrete for each fixed value of wavenumber and governing parameters and the eigenvalues s i (a; R), i ¼ 1, 2, . . . can be arranged in the order of decreasing real part. The basic flow is deemed linearly unstable if the real part of at least the first of the so-sorted eigenvalues satisfies the condition s R 1 max ; max a (s R 1 ) . 0, where the maximum is achieved at a ¼ a 1max . The value of the governing parameters R ¼ R c at which s R 1 max ¼ 0 is called critical. It corresponds to the bifurcation point at which the basic flow becomes linearly unstable. Of primary interest here is the 2p/a 1max periodic disturbance corresponding to the fastest growing small amplitude mode with s ¼ s 1 the temporal evolution of which we aim to investigate at the supercritical values of the governing parameter set R.
While the linearized consideration is capable of predicting the spatial shape of such a mode, its amplitude cannot be determined in the framework of linear analysis and this necessitates weakly nonlinear consideration. We outline it next for the situation when the basic flow becomes linearly unstable with respect to exactly one mode (and its complex conjugate), that is when In doing so, we will rely on the following two properties of the linear operator L a,s;R .
Remark 2.1. The leading eigenvalue s 1 of the linear operator L a,s;R has multiplicity 1 so that the rank of the N Â N operator L a,s;R is N 2 1.

Remark 2.2.
For any complex number s such that <{s} ¼ s R . s R 1 max . 0 operator L a,s is nonsingular.

Hierarchy of nonlinear terms and corresponding equations
Because of a quadratic nonlinearity of the governing equations, substitution of (2.4) in them produces terms that can be generally written as jAj 2 f 20 and A 2 f 22 E 2 þ c.c. They can only be balanced if (2.4) is extended to include functionally similar terms: It is easy to show that the additional terms must satisfy respectively. Since in (2.12) a ¼ a 1max and s ¼ s 1max , we set s R ¼ 2s R 1 max . s R 1 max . 0 in remark 2.2 and conclude that the operators in the left-hand sides of the above equations are non-singular and remain so even when R ! R c and s R ! 0. Therefore, unique solutions for w 20 and w 22 always exist.
Substituting (2.9) into the governing equations will produce yet another group of terms that are generally written as AjAj 2 f 31 E þ c.c. and A 3 f 33 E 3 þ c.c. in the equations. Balancing them requires a further extension of (2.9): In a way similar to that discussed above, we can show that a unique solution for w 33 is obtained from since the operator in the left-hand side of (2.15) is non-singular in view of remark 2.2 and remains so when R ! R c and s R ! 0. However, handling of equation for w 31 needs to be more delicate. It becomes If s R . 0, then in view of remark 2.2 the operator in the left-hand side of (2.17) is non-singular and the unique solution for w 31 can be found. However, if R ! R c and s R ! 0, L a,sþ2s R ;R ! L a,s;Rc and becomes singular so that the existence of solution w 31 cannot be guaranteed. Note that to this point no approximation has been introduced in the above procedure and all results obtained so far are exact. However, to resolve the potential unsolvability problem for w 31 the following approximation is required: it needs to be assumed that the evolution function F(A) can be represented in terms of an asymptotic series in amplitude that can be meaningfully truncated after a finite number of terms (of particular interest here is truncation after a cubic term in amplitude that leads to the so-called cubic Landau equation [20], where K is known as Landau constant). This requires the perturbation amplitude to remain small yet this does not impose any other explicit restrictions. In particular, it does not necessitate any explicit scaling restrictions on the parametric distance from a critical point (s R , R) ¼ (0, R c ) or, in fact, on the magnitude of s R . Consequently, as stated in [10] such an approach termed amplitude expansion can in principle be applied for a weakly nonlinear analysis even of linearly stable flows such as plane Couette or circular pipe Poiseuille flows for which a critical point cannot be found. In the limit of s R ! 0, the existence of the solution w 31 can only be ensured by enforcing the so-called solvability condition. Consider the adjoint eigenvalue problem where * denotes complex conjugation and the adjoint operator is defined with respect to some appropriately chosen inner product as hw y 11 , L a,s;R w 11 i ¼ hL y a,s;R w y 11 , w 11 i ¼ 0: Considering the inner product of the adjoint eigenfunction w y 11 with (2.19) in the limit s R ! 0, we obtain and, assuming that hw y 11 , Bw 11 i = 0, the Landau constant is uniquely defined using the chosen inner product as (2:23) The derivation above demonstrates that the definition of the Landau constant is unique at a critical point However, if s R = 0, the operator in the left-hand side of (2.19) is non-singular and the solution can be obtained for any value of K, which introduces the ambiguity at supercritical points with s R . 0 that are of main practical interest. Such an ambiguity was discussed in [10], where it was proposed to use an additional condition different to the above solvability condition to evaluate the Landau constant in supercritical regimes. However, the procedure was found to lead to numerical errors in the vicinity of the critical point because of the ill-conditioning of operator L a,sþ2s R ;R there. Subsequently, it was suggested in [19] to combine the tasks of finding w 31 and determining K into a single problem by adding an extra condition to the system. This was shown to work in practice, but no formal algorithmic proof was developed. In the next section, we present a generalization of the procedure for determining the Landau constant suggested in [19] along with a full proof of the algorithm that guarantees its uniform performance both at a critical point and a finite distance from it. We also prove that at a critical point the generalized procedure reduces to the application of a conventional solvability condition.

Evaluation of Landau constant
As follows from remark 2.2, system (2.19) is unconditionally solvable for any value of parameter K if s R . 0. To eliminate the ambiguity in defining the value of K, an additional condition is required. We choose to specify it in terms of the inner product where the superscript 'H' denotes Hermitian (conjugate) transpose and bold symbols represent N component complex-valued column vectors. The N Â N matrix operator M must satisfy the positivity condition for any complex valued vector w such that Mw = 0, but otherwise it is arbitrary. Combining (2.19) and (3.1) leads to the extended system of dimension N þ 1 First, we show that matrix operatorL is non-singular that is for any value of s R so that the unique solution can be always found. We consider cases s R = 0 and s R ¼ 0 separately. Taking into account that by construction L a,s;R w 11 ¼ 0, it is easy to check that this unique solution of (3.6a) is (3:7) By substituting (3.7) into (3.6b), we obtain (1=2s R )(w H 11 M) Á w 11 ¼ 0. Since Mw 11 = 0, this contradicts which is singular by construction. However, we will prove that detL ¼ Àhw y 11 , Bw 11 i = 0, (3:9) where w y 11 is the eigenfunction of the adjoint problem defined as Proof. Consider the co-factor matrix (3.11) of the matrix operatorL, whereĈ ij ¼ (À1) iþj det (L ij ) andL ij is a N Â N matrix formed by deleting the ith row and jth column fromL. The last element in (3.11) is zero because det L a,s;R ¼ 0 in Case 2. Introduce vector obtained from the last column in the matrixĈ (the framed part of (3.11)) by deleting the last element and compute detL by expanding by the last column inL Further, we expand the individual co-factorsĈ i(Nþ1) , i ¼ 1, . . ., N by the last row in each matrixL i(Nþ1) and obtain . . . is the co-factor matrix of matrix L a,s;R , C ij ¼ (À1) iþj det (L ij ) and L ij is a (N 2 1) Â (N 2 1) matrix obtained by deleting the ith row and the jth column in L a,s;R . This implies that Then upon right-multiplying by L a,s;R we obtain We show next that c = 0 and thus Recollect that matricesL i(Nþ1) consist of N 2 1 rows of L a,s;R {r 1 , r 2 , . . . , r iÀ1 , r iþ1 , . . . , r N }, and w H 11 M. Since det (L i(Nþ1) ) ¼ 0, the rows ofL ij are linearly dependent and there exist a set of scalar coefficients {a i1 , a i2 , . . . , a i(iÀ1) , a i(iþ1) , . . . , a iN , a i } that are not zero simultaneously such that . ., N, then any selection of N 2 1 rows from L a,s;R forms a linearly dependent set meaning that rank(L a,s;R ) , N À 1, which contradicts remark 2.1. Thus, the coefficient a i = 0 for some value of i in the range between 1 and N. Without loss of generality, it can be set to 1 to obtain where a T ¼ À(a i1 , a i2 , . . . , a i(iÀ1) , 0, a i(iþ1) , . . . , a iN Now we prove that hw y 11 , Bw 11 i = 0. To do so, we first demonstrate the validity of the following proposition.
Proof. Since rank(L a,s;R ) ¼ N À 1 (see remark 2.1), this matrix contains N 2 1 linearly independent column vectors. Then equation (3.16) becomes C T L a,s;R ¼ det L a,s;R I ¼ O, where O is a zero matrix of size N Â N. This means that the dimension of the null space of C T is n!N 2 1, or rank(C T ) ¼ N À n 1. Given that rank(L a,s;R ) ¼ N À 1 there exists at least one non-zero minor of L a,s;R of size N 2 1 so that the co-factor matrix C T of L a,s;R is non-zero and so is its rank. Thus, rank(C T ) ¼ 1. B Remark 3.4 (Full-rank factorization, see section 0.4.6 in [22] for details). Let C T be an N Â N matrix with complex elements. If rank(C T ) ¼ 1 then there exist two non-zero column vectors u and v such that Since v T = 0, L a,s;R u ¼ 0. Moreover, since u = 0 there exists a non-zero constant g 1 such that u ¼ g 1 w 11 . Similarly, from equation (3.16) we obtain Since u = 0, L y a,s;R v Ã ¼ 0 and since v = 0 there must exist a non-zero constant g 2 such that v Ã ¼ g 2 w y 11 . Therefore,  As stated in remark 2.1 the algebraic multiplicity of eigenvalue s is equal to 1. Thus the characteristic polynomial p(t) ¼ det L a,t is factored as where p 1 (s) = 0. Then it follows that: Hence hw y 11 , Bw 11 i = 0 and we finally conclude that detL = 0 when s R ¼ 0. B The main result of the above derivation is that a unique solution of (3.3) can always be found because the matrix operator in its left-hand side is non-singular. In particular, a unique (for a chosen weight matrix M) value of Landau coefficient can be obtained using Cramer's rule whereŜ is obtained fromL by replacing ÀBw 11 with f 31 for both cases s R = 0 and s R ¼ 0. Since the only difference betweenŜ andL is due to the replacement of the element ÀBw 11 with f 31 , the procedure identical to that used to derive (3.21) leads to detŜ ¼ hw y 11 , f 31 i, which can be formally obtained by replacing ÀBw 11 with f 31 in (3.21). Therefore, (3.26) becomes (2.23). Note that the adjoint eigenfunction w y 11 is defined up to an arbitrary multiplicative constant. Since we have shown that hw y 11 , Bw 11 i = 0 it can be chosen in such a way that Àhw y 11 , Bw 11 i ¼ 1. Then which is independent of M and is identical to the definition of Landau coefficient using a conventional solvability condition (2.23).  This condition was used in the work of Herbert [10]. In this case, the higher order in amplitude terms are chosen in such a way that the perturbation velocity component u 31 at point x i is set to zero so that in the vicinity of this particular point the perturbation of this specific velocity component is given completely by Au 11 (x i ). As was shown in [17], the results of such a projection depend on a (subjective) choice of the 'pinning' point x i . This inherently local procedure would also break if u 11 (x i ) and/or u 31 (x i ) happen to be zero so that the compulsory condition (3.2) is violated. (ii) In our earlier studies [18,23], M ¼ I was chosen that corresponds to the global orthogonality condition hw 11 , w 31 i ¼ 0. As was demonstrated in [17] such a choice removes the ambiguity associated with a subjectivity of choosing a 'pinning' point mentioned above. It also improves the convergence properties of the resulting asymptotic series. To see that rearrange (2.13) to read Asymptotically, the term in parentheses represents the fundamental disturbance harmonic given by an eigenfunction w 11 of the linearized problem and its higher-order distortion w 31 due to nonlinearity, with the leading term Aw 11 E being of primary interest. If hw 11 , w 31 i ¼ 0, the leadingorder approximation w ¼ w 00 þ Aw 11 E is not modified by the addition of higher order in amplitude terms as they are guaranteed to be orthogonal vectors. Essentially, (4.1) becomes an orthogonal expansion and its convergence properties improve. (iii) In [21], M was chosen to be the identity matrix I u,v , where the diagonal elements corresponding to pressure p were set to zero so that hw, Mwi ¼ u 2 þ v 2 , which has the meaning of kinetic energy. Such a choice preserves the benefits of the orthogonal expansion but produces the expansion that is optimal for capturing the kinetic energy of perturbations. This is useful when weakly nonlinear stability analysis results are compared with experimental flow data where velocity fields are measured directly while the pressure fields are not recorded (e.g. [24]). (iv) When integral spectral collocation methods are used as, for example, in [18,23], where the vector of unknown quantities is written in terms of a vector of their highest derivatives present in the governing equations multiplied by a spectral integration matrix (e.g. [25]), the diagonal identity matrices in the above examples become block-diagonal and so does matrix M. (v) The possibility of using a less conventional normalization condition for the disturbance amplitude was mentioned in [26] in the context of Rayleigh -Bénard problem: kv 11 , u 31 l ¼ 0, where the solution vector w ¼ (u, v, w, u, p) T consists of the velocity components u, v and w, temperature u and pressure p. While the physical meaning of such a normalization has not been discussed, formally it can be cast in the form of (3.1) with

An alternative condition
So far, a procedure for calculating Landau constants using condition (3.1) was discussed, in which the orthogonality of the higher-order perturbations with the eigenfunction w 11 or its selected components was used and interpreted from a physical point of view. The main result was that such a procedure automatically reduces to the use of a standard solvability condition involving the adjoint eigenfunction at a critical point. For completeness of the discussion, we should mention here that it is possible to formally apply this condition away from a bifurcation point enforcing as has been done in [12]. In this case, The proof thatL in expression (5.2) is non-singular is similar to that presented in §3 and is not repeated here. The meaning of the corresponding amplitude expansion in this case is that the full flow solution is projected onto a space spanned by w 00 and w y 11 . While such a projection is formally possible, its physical interpretation away from the critical point is not clear. In this context, the formulation considered in §3 that enables one to choose upfront a type of projection suited for particular practical purposes such as those listed in §4 is more flexible and thus may be preferred.

Illustrative example
In this section, we illustrate the numerical results obtained for various choices of M in the well-studied example of natural convection flow of fluid with Prandtl number Pr ¼ 7.5 contained between two vertical plates placed at non-dimensional positions x ¼ +1 and maintained at uniform non-dimensional temperatures T(+1)¼ +1. The sketch of a physical set-up and non-dimensional basic flow solutions are shown in figure 2. The solution vector in this case consists of four elements w T ¼ (u, v, T, p). Complemented with no-slip/no-penetration boundary conditions for velocities the system of equations (2.1) and (3.1) admits a steady basic flow solution of the form Such a simple flow becomes linearly unstable with respect to stationary y-periodic disturbances forming rolls with the axes in the mid-plane of the layer and wavenumber a c ¼ 1.383 at Gr c ¼ 491.8. Note that these values differ somewhat from a c ¼ 1.414 and Gr c ¼ 492.3 reported in [12] and are presumed to be more accurate given that they were obtained here using a spectral [25,27] rather than finite difference approximation. Because the developing instability is stationary, the linear amplification rate remains real (s ¼ s R ) and so does Landau constant K ( figure 3a,b). In supercritical regimes, the wavenumber corresponding to the largest value of s R decreases and s R increases mostly linearly with Grashof number. Consistent with the proof given in §3 the use of all weight matrices M leads to the same value of Landau constant at the bifurcation point. This value K ¼ 211.632 is identical to that obtained from a solvability condition (2.23) involving the adjoint eigenfunction. This value is also close to the one obtained using the correlation formulae K ¼ À720(Pr 1:325 =Gr c a 2 c ) % À11:050 suggested in [15]. The values of K ¼ 210.102 and K ¼ 26.153 that we calculated following the procedure outlined in §5 (see the thin solid lines in figure 3) by fixing a 0 ¼ 1.414 for supercritical values of Gr ¼ 500 and Gr ¼ 550 as was done in [12] agree with their values of K ¼ 210.100 and K ¼ 26.151 (obtained upon rescaling the values given in Table I in [12] as K ! K=(Gra 2 0 ), which results from a different normalization of the eigenfunctions of a linearized problem used here).
As expected, in the close vicinity of the critical points all projections based on various weighted inner products produce numerical estimations of the perturbation amplitude that are virtually indistinguishable from each other (figure 3d). However, they all differ noticeably from the amplitude predictions based on Landau constant evaluated at a critical point shown by the thick dashed-dotted line. This is due to an inherent inconsistency of the way such a prediction is obtained. While its core assumption is that K remains constant for supercritical values of Gr, see the thick dashed-dotted line in figure 3b, to estimate non-trivial values of the amplitude jA e j ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Às R =K p one has to use the actual supercritical values of s R computed for Gr . Gr c as shown in figure 3a.
Further away from a critical point the amplitude estimations based on different weighted products produce numerically different results. Again, this is expected because the 'effective dimensionality' of the full problem solution increases, see the oval region in figure 1, while by its very nature each lowdimensional projection can emphasize only a limited subset of full flow features. Therefore, it becomes important to specify which solution characteristics are of the main interest in a particular physical context and embed this focus in the projection procedure from the outset. For example, in studies of isothermal fluid flows the dynamic characteristics of a perturbed flow are of primary interest [24] so that the projection selecting the kinetic energy [21] could be preferred, see Case 3 in §4. On the other hand, the main practical interest in convection problems such as the one considered in this section is where the temperature derivative is evaluated at one of the walls at x ¼ +1. Therefore, the use of the weight matrix that selects the perturbation temperature and downplays the other components of the solution may be preferred. The behaviour of the so-computed Nusselt number in supercritical regimes is demonstrated in figure 4a by the dotted line. It is virtually indistinguishable from that obtained by following the procedure outlined in §5 and previously used in [12] near the critical point, but the values obtained using these two approaches deviate somewhat further away from it. The dashed line depicts results obtained using a weight matrix emphasizing kinetic energy of the perturbation. They are close to but not identical to predictions based on (6.3).
To compare the Nusselt number estimations with DNS and experimental data available in the literature, we also computed the results for Pr ¼ 0.71. They are shown in figure 4b. As expected, the low-dimensional projection results underestimate those of direct numerical simulations [29] and experiments [28] because they only take into account one fastest growing Fourier component with a fixed wavenumber while Fourier components covering a finite wavenumber range are present in supercritical regimes (see also

Conclusion
In this study, we revisited the procedure of amplitude expansion in the context of weakly nonlinear stability theory of flows arising in extended domains with at least one finite dimension. We emphasized the application of this procedure at parametric points located finite distance away from the critical point where the real amplification rate of infinitesimal disturbances becomes zero. We confirmed that the definition of the perturbation amplitude in this case is not unique and its meaning has to be specified from the outset to obtain meaningful physical interpretation of results. We demonstrated that this can be done by introducing an appropriate orthogonality condition with respect to a weighted inner product. The non-uniqueness of the amplitude definition offers an opportunity for choosing it in such a way that the resulting low-dimensional projection of the full solution emphasizes its specific features (e.g. kinetic energy or heat flux) that are of interest in a particular physical context. The current procedure is a generalization of those previously suggested in [10,12,18] and contains them as special cases. The main outcome of the current work is that the computational procedure for evaluating Landau constants in supercritical regimes finite distance away from a critical point, the idea of which was initially formulated in [19], has been put on a firm ground by formal proof of the facts that it does not result in any singularities and automatically recovers standard solvability condition at a critical point. While the work has focused on the evaluation of the first Landau constant appearing at the third order of amplitude, exactly the same procedure can be uniformly applied to evaluate Landau constants at higher orders of disturbance amplitude. Finally, we note that the procedure for evaluating Landau constants discussed here is also applicable for amplitude expansions in subcritical regimes (e.g. when basic flow is subject to a subcritical bifurcation at a critical point or when the basic flow remains stable with respect to  shows experimental results of [28]. The diamonds correspond to direct numerical simulation of [29]. infinitesimal disturbances). However, in this case special care needs to be taken in treating subcritical resonances that occur between decaying instability modes, see [3,11,18]. It follows from the presented proof that such resonances cannot arise in supercritical systems, but one has to be mindful of them in subcritical regimes because L a,sþ2s R ;R can in principle become singular if s R , 0. If this occurs, a system of coupled amplitude equations accounting for the resonant mode interaction needs to be considered, as discussed in [21]. A systematic procedure of resolving second and higher-order subcritical resonances suggested there can be used to derive a system of coupled Landau equations modelling the evolution of resonating subcritical modes. The computational procedure formulated here does not require any modifications to be used for evaluating Landau coefficients of such a system.
Data accessibility. This article has no additional data. Authors' contributions. S.A.S. conceived the mathematical concept and provided its interpretation, K.G.P. developed mathematical proofs and obtained computational results. Both authors wrote the paper and gave the final approval for publication.
Competing interests. We declare we have no competing interests. Funding. K.G.P. acknowledges financial support the Ministry of Education and Training of Vietnam in the form of the VIED-MOET PhD scholarship.