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[MATH] the two stable fixed points of the system, for the given values of [MATH] and [MATH] , become unstable. The value [MATH] marks the threshold of sustained chaotic behaviour, on which we focus, for almost all initial conditions, in contrast to the transient chaos that can be observed on an unstable chaotic set for... |
(Yorke & Yorke, 1979 . The oscillations in [MATH] in figure are due to the fact that [MATH] is an estimator obtained from a finite time interval. Indeed, comparison of the statistic obtained over |
[MATH] with that obtained over [MATH] in figure , indicates that the oscillations reduce in amplitude as the length of the time interval increases. |
2.1 A finite difference approach A finite difference approach An estimation of the Gâteaux derivative of a single functional [MATH] with respect to the [MATH] th component [MATH] of [MATH] is |
[EQUATION] in which all elements of [MATH] are equal to zero, with the exception of the [MATH] th element, which is equal to [MATH] . If [MATH] is relatively large, then |
[MATH] will not provide an accurate approximation to the local derivative. If, on the other hand, [MATH] is relatively small , the non-smooth behaviour of |
[MATH] for finite-time averages evident in figure suggests that we would need to obtain statistics over a correspondingly large time to obtain meaningful results |
(see e.g. Russo & Luchini, 2016 . Moreover, the use of such an approach to obtain the sensitivity of [MATH] with respect to other system parameters requires the entire simulation to be run at least twice for each parameter [MATH] |
2.2 The tangent linear equations The tangent linear equations Instead of by looking at finite differences between independent simulations, a functional’s gradient can, in theory, be calculated exactly using information from a single simulation. We will outline a naive version of the method before explaining its problem... |
[EQUATION] where [MATH] . By augmenting the time-dependent functions with boundary values: [EQUATION] and defining an inner product [MATH] , based on ), for the space to which the elements |
[MATH] and [MATH] belong, the functional’s derivative can be expressed as (see e.g. Sewell, 1987 [EQUATION] In general, a penalty term that depends on the state trajectory’s end point [MATH] can be added to ), which would modify the [MATH] that appears in ( for [MATH] . The perturbation [MATH] |
satisfies the tangent linear equations, which are obtained by differentiating [MATH] with respect to [MATH] [EQUATION] where [MATH] sets the perturbation of the initial condition to be zero. For the Lorenz equations the tangent linear system for [MATH] is equivalent to |
[EQUATION] in addition to the initial condition [MATH] . Note that the [MATH] [MATH] and [MATH] appearing in ) are known, albeit time-dependent, variables. Once the perturbed trajectory |
[MATH] is known, the derivative of any functional [MATH] with respect to [MATH] can in principle be calculated using ( ) by changing |
[MATH] . Calculation of the derivative of a given [MATH] with respect to a different parameter [MATH] is more difficult, because it requires us to find a different perturbed trajectory from ( ) for use in ( ). This motivates an alternative way of factorising ( ), to obtain adjoint variables that describe the change in ... |
2.3 The adjoint equations The adjoint equations Introducing the adjoint variables [MATH] to enforce the the equations of motion [MATH] , which act as constraints, results in |
[EQUATION] where [MATH] is an inner product for the dual space containing the elements [EQUATION] If [MATH] satisfies the adjoint equations: |
[EQUATION] where [MATH] denotes the adjoint/transpose of an operator, then integration by parts of ( 10 ) results in [EQUATION] Note that the penalty term [MATH] in |
[MATH] corresponds to the initial condition [MATH] , and that the penalty term on [MATH] in ( 13 ) corresponds to the initial condition [MATH] . Since [MATH] |
is equal to the derivative of [MATH] with respect to a change in the constraints, equation ( 13 ) enables us to readily compute the sensitivity of a given functional [MATH] |
with respect to any parameter [MATH] . To appreciate this, observe that ( 13 ) contains [MATH] , whereas 12 ) does not, and that the converse statement is true for ( ) and ( ). On the other hand, calculation of the the sensitivity of a different functional is difficult using the adjoint approach, because it would requi... |
Solutions to both the tangent system ( ) and the adjoint system ( 12 ) grow without bound as the time [MATH] in ( ) increases. As pointed out by |
Thuburn ( 2005 , the cause of the difference between the actual gradient and a gradient obtained from either the tangent or adjoint system is the fact that the operation of time averaging over [MATH] does not, in general, commute with the finite difference of an infinite time average over an interval of [MATH] |
[EQUATION] The finite difference of the functional [MATH] does not converge uniformly to the sought-after derivative for all integration times [MATH] and, therefore, neither do ) nor ( 13 ). |
An approximation to [MATH] can be obtained if 12 ) is integrated over relatively short time intervals (Lea et al. 2000 . On the other hand, if finite differences are employed using equation ( ), then the minimal time [MATH] |
over which accurate statistics can be obtained is determined by [MATH] . An accurate finite difference approximation requires a small value of [MATH] , which requires a large value of [MATH] |
(see e.g. Russo & Luchini, 2016 . Thus, approximate gradients can be obtained by using the tangent linear equations or finite differences, provided that small or large time intervals are used, respectively. |
That it is crucial to take the limit [MATH] before analysing derivatives suggests that sensitivity analysis of the equations governing the statistics of the process might result in a more tractable problem. At the expense of introducing additional unknowns, we therefore focus on obtaining adjoint information for the eq... |
3 The cumulant equations and their closure The cumulant equations and their closure The equations that govern the behaviour of cumulants provide a means of establishing the leading-order relationships between the statistics of a chaotic attractor. These relationships constrain the response of statistics to changes in p... |
3.1 The cumulant generating functional The cumulant generating functional The cumulants [MATH] of a dynamical system can be defined in terms of a cumulant generating functional [MATH] |
[EQUATION] where [MATH] is a multi-index for the system of [MATH] time-dependent variables [MATH] , and [EQUATION] where [MATH] is the Hopf generating functional |
(see e.g. Hopf, 1952 ; Frisch, 1995 and [MATH] . The over-bar in ( 15 denotes the finite time average defined in ( ), which we assume converges to a phase average when the duration the averaging interval [MATH] . Due to the logarithm in ( 15 ), a cumulant, unlike a moment, derived from the sum of two independent random... |
[EQUATION] where the coefficients of the three terms on the right-hand side correspond to the number of ways of partitioning a multiset of three (identical) elements into (a) a single multiset of three; (b) a multiset of two and a set of one; (c) three sets of one. In this respect, cumulants are the atoms of which mome... |
[MATH] , and [EQUATION] in which the grouped terms correspond to a summation over the different ways that the multiset of four elements |
[MATH] can be partitioned into subsets of a given cardinality. The connection between cumulants and moments is discussed in more detail in appendix , in which it is helpful to compare ( 18 ) with ( 47 ). |
Noting from ( 16 ) that [MATH] plays the role of [MATH] , the Hopf function [MATH] satisfies the linear equation [EQUATION] The original [MATH] nonlinear equations [MATH] from ) are recovered by differentiating ( 19 with respect to the vector [MATH] . Associated with the original system ( ) are an infinite hierarchy of... |
[EQUATION] which are obtained, under the assumption of ergodicity, by averaging ( 19 ) to obtain a stationary equation for [MATH] , in which the equation for a given cumulant [MATH] corresponds to the coefficient of [MATH] . Readers are referred to |
Frisch ( 1995 for further details. Despite the fact that they do not form a closed system, the cumulant equations provide useful information. For example, as noted by Knobloch ( 1979 , the cumulant equations for the Lorenz system indicate that |
[EQUATION] which, since [MATH] , implies that [MATH] in a statistically steady state. If the original system evolves on a [MATH] -dimensional phase space then, ignoring symmetries in the governing equations, the number of cumulants at order [MATH] is equal to the number of ways that [MATH] |
indistinguishable objects can be assigned to [MATH] sets; hence the number of cumulants up to and including those of order [MATH] is |
[EQUATION] as illustrated in figure [MATH] . Known symmetries of a system reduce the number of independent unknown cumulants. In the case of the Lorenz equations ), for which [MATH] is invariant under the mapping [MATH] , a cumulant |
[MATH] for which [MATH] is odd, is necessarily equal to zero. In the case of [MATH] , of the [MATH] and [MATH] available cumulants up to order [MATH] and [MATH] [MATH] and [MATH] , respectively, are non zero. Closures of the cumulant hierarchy aim to strike a balance between the incorporation of additional physics from... |
3.2 Adjoint cumulant dynamics Adjoint cumulant dynamics As described in § , if one wishes to differentiate a vector-valued function with respect to vector-valued input one can employ one of two dual approaches. Each approach uses the chain rule: one forwards and the other backwards. The first approach is to propagate d... |
Regarding [MATH] , originally defined in ), as a functional of an infinite hierarchy of cumulants, [EQUATION] where the cumulant perturbation [MATH] and the corresponding adjoint variables [MATH] are defined according to |
[EQUATION] As discussed in § , if the problem involves more functionals than parameters, it is computationally preferable to solve the tangent linear system to find [MATH] before evaluating ( 23 ). If, on the other hand, the problem contains more parameters than functionals, then it is computationally preferable to fin... |
[EQUATION] where [EQUATION] We focus on the adjoint problem of determining the sensitivity of a single functional [MATH] (we omit the subscript |
[MATH] hereafter) with respect to a potentially large number of unknown parameters. Unlike the systems ( ) and 12 ), whose validity relies on the commutation of time averaging and differentiation with respect to [MATH] 23 ) works with time averaged variables directly. |
3.3 Building the cumulant operator Building the cumulant operator If the original system of equations ( contains nonlinear terms then the equations for the cumulants of order [MATH] will depend on cumulants of order [MATH] and higher, depending on the degree of nonlinearity. For the Lorenz equations ), and indeed the q... |
[EQUATION] Here [MATH] represents perturbations [MATH] to the cumulants of order [MATH] . According to ( 22 ), each operator [MATH] has the shape |
[EQUATION] For the Lorenz equations, [EQUATION] Whilst the tangent linear system is under determined , the adjoint system [MATH] is |
over determined . The overall properties of the system can be be seen in the self-adjoint problem that combines the tangent linear and adjoint operators. With the equations for perturbations to the first order cumulants (e.g. [MATH] [MATH] and |
[MATH] ), one finds [EQUATION] If the functional [MATH] , and therefore the vector [MATH] , is specified then one can solve for the adjoint variables |
[MATH] according to the second row of 30 ). However, a consistency requirement for the extended system ( 30 ) to possess a solution is that |
[MATH] , in general. We are therefore not at liberty to choose the functional [MATH] arbitrarily, because it will automatically contain a contribution scaled by [MATH] from the unclosed perturbations [MATH] |
The vacuous consequence of using ( 30 ) is that only functionals whose value can be determined identically from the original cumulant equations, such as equation ( 21 ), can be determined exactly. For ( 30 ) to yield novel information an assumption is required about the response of the unknown cumulant perturbations [M... |
[MATH] , which corresponds to the unknown high-order perturbations [MATH] being either zero or orthogonal to the weighting vector |
[MATH] , as illustrated in figure [MATH] More generally, taking the system ( 30 ) as an example, a closure corresponds to the specification of |
[MATH] in terms of the the retained cumulant sensitivities [MATH] . If [MATH] belongs to a three-dimensional parameter space, then truncation at order [MATH] entails three assumptions, determining [MATH] for |
[MATH] . In this respect, the number of required assumptions is independent of the order [MATH] at which a closure is invoked, which arguably makes finding a suitable closure for sensitivity analysis less onerous than finding a suitable closure for the original cumulant equations. |
3.4 Closure Closure As illustrated in figure , to obtain a closed system of cumulant equations one needs to make an assumption about how the highest-order cumulants are related to those of lower order and, therefore, the way in which they depend on the problem’s parameters. One approach is to assume that cumulants whos... |
[MATH] . For [MATH] in a system with quadratic nonlinearities, this approach is consistent with the assumption that the probability distribution of the underlying process is Gaussian and is therefore completely determined by its cumulants of first and second order |
(Frisch, 1995 As outlined in section § 3.3 , for sensitivity analysis the implications of discarding cumulants beyond a certain order are weaker than those associated with direct simulation of the truncated equations. For example, truncation of the cumulant equations at order [MATH] and assuming that |
[MATH] , produces non-realisable statistics (Kraichnan, 1980 , leading to a negative energy spectrum in turbulence (Ogura & Phillips, 1962 . From the perspective of sensitivity analysis, however, the orthogonality condition |
[MATH] does not necessarily imply that [MATH] . Similarly, [MATH] does not necessarily imply that the process is Gaussian. It is nevertheless important to note that the behaviour of higher-order cumulants in a Gaussian distribution is a special case, because probability distributions possessing non-zero cumulants at or... |
[MATH] , followed by zero cumulants at all orders [MATH] do not exist (Lukacs, 1970 , p. 223) One can discard cumulants of order higher than [MATH] and model their effects with a forcing function such as [MATH] which, in general, will depend on a vector [MATH] of unknown parameters: |
[EQUATION] where [MATH] represents the residuals arising from the truncation and [MATH] is the projection that sets the value of cumulants whose order exceeds |
[MATH] to zero. Assuming that the residual [MATH] can be made small with a suitable choice of [MATH] , and that for a given [MATH] [MATH] does not depend on [MATH] , the tangent linear equations at order [MATH] are |
[EQUATION] A key assumption underlying the use of 32 ) as a model for the tangent linear behaviour of the system is that the model parameters [MATH] in 31 ) exhibit a weak dependence on the problem parameters [MATH] (hence |
[MATH] is not included in 32 )), which is consistent with the assumption that [MATH] in the vicinity of [MATH] . Utilising ( 32 ) for truncation at |
[MATH] in the sensitivity analysis of a system with quadratic nonlinearities, under the assumption that [MATH] depends only on the highest retained cumulants |
[MATH] , yields [EQUATION] where [EQUATION] The closed system of extended equations 33 ) is, in general, invertible and therefore provides a set of solutions for the adjoint variables |
[MATH] for a specified set of weights [MATH] . Without selecting the model parameters [MATH] , inversion of the adjoint operator |
[MATH] yields a fan of gradients, as indicated in figure [MATH] . The determination of a unique gradient from the fan requires the selection of an optimal set of model parameters [MATH] . For example, the optimal parameters could be chosen to minimise |
[MATH] according to statistical observations from a direct simulation: [EQUATION] Using local observational data the procedure of obtaining sensitivity information can therefore be freed from tunable parameters once a suitable class of models has been selected. |
The extent to which it is necessary to include cumulants of order greater than [MATH] for sensitivity calculations depends on the role they play in maintaining the statistical equilibrium defined by ( 20 ). Although the truncation of the cumulants at second order yields realisable statistics, the second order cumulants... |
(Bohr et al. 2005 . Therefore, a popular choice, known as the Eddy Damped Quasi-Normal Markovian approximation (see e.g. Leith & Kraichnan, 1972 , is to truncate the cumulants at third order and to include a damping term to account for the discarded fourth-order cumulants: |
[EQUATION] When [MATH] the cumulants of order [MATH] become increasingly damped and the closure corresponds to a truncation at [MATH] ; when [MATH] the closure corresponds to truncation at [MATH] |
(Allawala & Marston, 2016 . The eddy-damping parameter therefore produces a fan of possible functional gradients, as illustrated in figure [MATH] |
The optimal value of [MATH] that minimises the size of the normed residual [MATH] is [EQUATION] which enables the optimal functional gradient to be determined according to |
[EQUATION] The procedure described in this section consists of identifying the order [MATH] at which the cumulant hierarchy should be truncated, before selecting a subclass of possible models for the unknown cumulants. The optimal parameters [MATH] , and therefore optimal gradient [MATH] in figure |
[MATH] , can be determined by minimising the residual between statistics from direct simulation and the corresponding model prediction according to equation ( 35 ). |
4 Two-dimensional convection (the Lorenz equations, [MATH] ) Two-dimensional convection (the Lorenz equations, [MATH] 4.1 Truncation of the cumulant hierarchy Truncation of the cumulant hierarchy |
To test the method for obtaining functional gradients described in 3.4 , we collect statistics from direct simulations of the Lorenz equations ( ). We focus on the statistically stationary state produced by parameters |
[MATH] , which is well documented and was the state chosen for the sensitivity analysis of Wang ( 2013 . The dynamical equations are integrated using the DOPRI5 explicit Runge-Kutta method in Python’s SciPy library. To check convergence of the computed cumulants the time [MATH] used to define the time average ( ) was v... |
[MATH] to [MATH] . To allow for transient behaviour, the initial time used in the simulations is [MATH] . Integrals such as ( ) were computed using a trapezium rule over the discrete points obtained from the simulations. |
Gradients of the functional [MATH] are displayed in figure , which shows the projection of the gradient vector [MATH] onto two-dimensional planes. The symbols denote the gradients that are obtained by truncating the cumulant hierarchy at order [MATH] , without modelling the discarded cumulants. Truncation of the tangen... |
[MATH] yields an inaccurate representation of the gradient of [MATH] . The response of the second order cumulants to changes in the parameters is evidently significant and therefore the assumption that their dependence on parameters is identically zero (or, more generally, orthogonal to [MATH] , as described in § 3.3 )... |
[MATH] , but one that is an improvement in comparison with truncation at [MATH] . As discussed in 3.4 , in shearless turbulence the effect on eddies of eddy-eddy interactions, captured by the third order cumulants |
(Farrell & Ioannou, 2014 , is expected to play a crucial role in maintaining statistical equilibrium in the case of the Lorenz equations. Indeed, the third order cumulants play a dynamically important role in determining the response of the Lorenz system to parametric changes, and figure shows that their retention yiel... |
[MATH] Truncation of the cumulant equations at [MATH] yields a poor approximation to [MATH] , which illustrates the need to find a compromise between the efficiency and simplicity of truncation at relatively low order and the additional physics that is captured by higher-order cumulants. In the absence of physical just... |
is inaccurate. 4.2 Error analysis Error analysis The difference between the approximation [MATH] and the observed gradient [MATH] that was obtained by truncating the cumulant hierarchy (depicted in figure with a star) can be understood by inspecting the derivatives of the discarded cumulants. As discussed at the end of... |
[MATH] , where [MATH] are the perturbations of the neglected cumulants, and [MATH] determines the influence they have on the functional in question: |
[EQUATION] We focus on the error associated with the derivative of [MATH] with respect to [MATH] (i.e. [MATH] ), and display [MATH] and [MATH] for [MATH] |
and [MATH] in figures and . We restrict attention to non-zero cumulants using the symmetry arguments made in § 3.1 . The gradients were determined by analysing statistics from [MATH] simulations employing values of [MATH] |
distributed uniformly over a unit interval centred on [MATH] . Further details are provided in appendix The error associated with truncation at [MATH] in figure |
is entirely due to the behaviour of the cumulant [MATH] . The remaining cumulants, for which the corresponding values of [MATH] are zero, do not contribute to the error, as can be seen directly from the second row of the cumulant equations in equation ( 29 ). At order [MATH] in figure , the dominant contribution to the... |
[MATH] , shown in figure , it comes predominantly from [MATH] and [MATH] , which are related to the moments [MATH] and [MATH] . For truncation at order [MATH] the perturbations in the discarded cumulants are large |
[MATH] , with figure indicating that the dominant contribution to the error comes from [MATH] , which is related to the moment [MATH] . The effect on the error of the growing sensitivity and number of discarded cumulants is, to a limited extent, compensated by their diminishing influence on the gradient [MATH] , as evi... |
A summary of the truncation errors obtained at each order is provided in table . Obtaining accurate observations of the sensitivity of fifth-order statistics from the Lorenz attractor is challenging, because it requires the use of relatively large intervals for time averaging. The approximate equality between the third... |
[MATH] and the error [MATH] agrees with [MATH] , and therefore satisfies the original cumulant equations to within [MATH] . At orders [MATH] and [MATH] the difference between the third and fourth columns of table |
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