We propose several continuous data assimilation (downscaling) algorithms based on feedback control for the 2D magnetohydrodynamic (MHD) equations. We show that for sufficiently large choices of the control parameter and resolution and assuming that the observed data is error-free, the solution of the controlled system converges exponentially (in and norms) to the reference solution independently of the initial data chosen for the controlled system. Furthermore, we show that a similar result holds when controls are placed only on the horizontal (or vertical) variables, or on a single Elsässer variable, under more restrictive conditions on the control parameter and resolution. Finally, using the data assimilation system, we show the existence of abridged determining modes, nodes and volume elements.
In the study of solar storms, space weather forecasting, earth’s geodynamo, and other areas, predicting the motion of fluids with magnetic properties is a central concern. The governing equations are often taken to be the magnetohydrodynamic (MHD) equations, or some modification of them. These equations are notoriously difficult to solve both analytically and computationally. Moreover, accurately initializing the system is challenging due to the sparsity of the available data. Fortunately, data is often given not just at a single time, but can be streaming in (e.g., from devices monitoring space plasma dynamics), or given in history (e.g., from surface geomagnetic observations, which in the earth can be traced back up to 7000 years [8,16,55]). This situation is similar to the problem of weather prediction on earth. Therefore the techniques of data assimilation, which were developed in weather prediction, have been applied to the MHD equations in recent years (see, e.g., [11,13,33,34,38,49,52,56,58,59]). It has also been speculated in [1] that data assimilation for magnetohydrodynamics may be useful in liquid sodium experiments modeling the Earth’s core.
Data assimilation has been the subject of a very large body of work. Classically, these techniques are based on linear quadratic estimation, also known as the Kalman Filter. The Kalman Filter has the drawback of assuming that the underlying system and any corresponding observation models are linear. It also assumes that measurement noise is Gaussian distributed. This has been somewhat corrected via modifications, such as the Extended Kalman Filter and the Unscented Kalman Filter. For more about the Kalman Filter and its modifications, see, e.g., [18,45,48], and the references therein. Recently, a promising new approach to data assimilation was pioneered by Azouani, Olson, and Titi in [4,5] (see also [12,39,54] for early ideas in this direction). This new approach is based on feedback control at the PDE level. The first works in this area assumed noise-free observations, but [6] adapted the method to the case of noisy data, and [29] adapted it to the case where measurements are obtained discretely in time and may be contaminated by systematic errors. Computational experiments on this technique were carried out in the cases of the 2D Navier–Stokes equations [36], the 2D Bénard convection equations [3], and the 1D Kuramoto–Sivashinsky equations [47,50]. In [47], several nonlinear versions of this approach were proposed and studied. In addition to the results discussed here, a large amount of recent literature has built upon this idea; see, e.g., [2,7,22–26,35,37,40,41,46,51,53].
In the present work, we adapt the approach of [4,5,23] to the 2D MHD equations. In Theorem 3.1, we show that solutions of the feedback-controlled system converge exponentially in the -norm to solutions of the MHD system when feedback control is applied to all variables (here, we use Elsässer variables for simplicity). This convergence holds under certain conditions on the spacing of the data and the weight given to the feedback control. Moreover, in Theorems 3.2 and 3.3, we establish abridged data assimilation, i.e., we show that feedback control need only be applied to a reduced set of the variables (horizontal variables or a single Elsässer variable, respectively) to obtain exponential convergence, at the cost of more restrictive conditions on the data resolution h and control weight μ. In Theorem 3.4, we establish exponential convergence in the -norm. Next, in Theorem 3.8, we show that if one makes weaker assumptions on the data interpolation function, and if feedback control is applied only to horizontal variables, then exponential convergence in the norm holds as well. Finally, in Section 3.3, we establish a rigorous connection between data assimilation and the concept of determining quantities, first introduced in [30], and further studied in [15,31,42–44].
Background on data assimilation
We now describe the general idea of the data assimilation scheme we use for the 2D MHD equations, based on the idea of feedback control, that was developed by Azouni, Olson and Titi in [4,5] in the context of the 2D Navier–Stokes equations. In the study of a dynamical system in the form,
subject to certain boundary conditions, one normally tries to show that unique solutions will arise given any initial value
in a certain space, and that the solution will change in a continuous way with respect to a change in the initial value.
The problem arises in practice that the initial value may not be known exactly, but it may approximate the true initial value of a given observable, for example the temperature, which we’d like to predict the value of in the future. The continuous dependence on initial data addresses this issue, in that if the initial approximation is close enough to the true value, then the solution we obtain will accurately approximate the true value of the observable for some period of time. However, usual theory shows that the length of time the approximation is guaranteed to be good is short, in that the error may grow exponentially in time. Also, the initial measurement may need to give a very close approximation to the true initial value, but in practice measurements may only be available on a coarse grid, limiting the accuracy of the initial approximation and thus limiting both the accuracy the solution can be guaranteed to have, as well as the duration for which this accuracy can be guaranteed.
Data assimilation is the method where, to compensate for this lower bound on the accuracy of the measured initial condition, measurements are taken of the observable as time goes on (over the same possibly coarse grid on which the initial value is approximated) and fed back into the differential equation (giving a different equation, called the data assimilation equation) in such a way that the solution will become a better approximation as time goes on. This gives us the accuracy we need to apply the continuous dependence on initial data and say the prediction will be accurate for some duration from that time onwards.
The data assimilation algorithm (the way measurements are introduced to the differential equation) can take different forms, but the one we consider here was first introduced by Azouani, Olson, and Titi in [4,5]. Given that the true value of the observable at time t is then the data assimilation equation will be:
where the second equality in the above equation follows because we’ll assume the interpolant operator, , is linear. Here, μ will be an adequately chosen tuning parameter. In addition, we will assume that for all , satisfies one of the following:
or
Many relevant examples of operators satisfy one of these two conditions, including the projection onto the low modes, finite volume element operators, and nodal interpolant operators. For more information, see, e.g. [4,32,50].
Background on the MHD equations
We consider the 2D MHD equations for a fluid and magnetic field under periodic boundary conditions and with zero space average. Let , , and p represent the fluid velocity, magnetic field, and fluid pressure, respectively, and let the spatial domain be . The system can be written as (see, e.g., [19]):
Here, is the kinematic fluid viscosity, is the fluid density, is the permeability of free space, is the magnetic diffusivity, and σ is the electrical conductivity of the fluid. We impose initial conditions and in an appropriate function space, and allow for time-dependent forcing functions, denoted above by f and g.
Our analyses will have to take into account the amount of energy being added to the system by the forcing functions, so to this end we define the Grashof number, G, to be
where is the smallest eigenvalue of the Stokes operator on the space of functions with space average zero on under periodic boundary conditions [27].
Note that we have constructed G to be dimensionless. We will also non-dimensionalize the system so that we can later reformulate it in terms of the Elsässer variables. Let U be a reference velocity and use L as a reference length. We denote the dimensionless fluid Reynolds number and the dimensionless magnetic Reynolds number by and , respectively. In non-dimensional form, the system can be written as:
with the initial conditions and , and where is the (non-dimensionalized) sum of the fluid and magnetic pressures, and , , , , f, and g have been replaced by their appropriate non-dimensional versions. Note the bilinearity in on the left-hand side of (3b) allows for the important fact that the four non-linear terms in (3) can be written with coefficients . We will denote the non-dimensionalized spatial domain by
Global existence and uniqueness of solutions to (3) was proven in [20] and [57]. Recent work on connection to magnetohydrdynamic turbulence can be found in [9,10]. For a derivation and physical discussion of the MHD equations, see, e.g., [14]. For an overview of the classical and recent mathematical results pertaining to the MHD equations, see, e.g., [19,20].
Preliminaries
In this section, we briefly lay out some notation, discuss some of the standard results and inequalities we use, and give the specific equations we will discuss.
For a matrix A, we denote . We denote the standard inner-product and norm by and , respectively (note that the integral is taken over the non-dimensionalized domain, Ω, so has the same units as u). We also denote , which is equivalent to the standard norm, due to the Poincaré inequality (6).
We recall some standard inequalities. Here , , and and w are divergence-free periodic functions, with sufficient regularity to make all the norms involved finite.
We will frequently use the following forms of Young’s inequality and Hölder’s inequality:
We also recall the following version of Poincaré’s inequality, valid for periodic functions with zero space average on Ω:
The following inequality due to Ladyzhenskaya will be used to bound the nonlinear terms for the cases where we have measurements on all the components and when we only measure one Elsässer variable:
The next two inequalities are extensions of the Brezis–Gallouet and are due to Titi [61]. They will be necessary to bound the nonlinear terms in the case of measuring only one component of the reference velocity and magnetic fields:
where in (9), z can be u or v.
The following generalization of the Grönwall Lemma will be useful, which was first shown by Foias et al. in [28]. For a proof of an even more general version due to Jones and Titi, see [27].
(Generalized Gronwall Inequality).
Letbe a locally integrable function such that for somethe following two conditions hold:whereThen ifis absolutely continuous and for almost allwhereas, thenas well. Furthermore, ifthenexponentially as
Next, in order to simplify our calculations we will reformulate the MHD equations in terms of new variables which we call and , in such a way as to symmetrize the system.
We assume, without loss of generality, that , and denote the Elsässer variables [21] by and (if then we would denote and proceed similarly).
Then we can derive evolution equations for and by considering both the sum and difference of (3a) and (3b) and obtain the following system:
subject to the initial conditions and .
Here we relabeled the forcing terms as and , and we denote and . It will be important to note that and that and (this last inequality is true by the assumption that , however if then we would arrive at the above system except with a different sign on the pressure, and , so still we have , and in general we will have ).
We note here that G can be expressed in terms of the forcing functions for the reformulated system:
hence,
Now, we describe the data assimilation algorithms studied in this paper. Following the ideas of [4,5] we incorporate measurements obtained from a fixed reference solution (of which we want to predict future values) through a damping term. This will “steer” the data assimilation solutions to the reference solution exponentially in time. In what sense we will have convergence depends on the type of interpolant with which we take measurements.
The results are separated by the type of interpolant considered and by which measurements are recorded. We frame our results in terms of the Elsässer variables, not in terms of and . Also, we consider algorithms which require measurements taken only on the first components, and (which is the same as measuring and ), by measuring all the components of and , or by measuring either the sum or the difference only.
In the following, let be a fixed solution of (11), and we denote the data assimilation variables by and , which will approximate and respectively. may satisfy either (1) or (2), and we will analyze each case separately. Because we are introducing the feedback term into the equations, the magnetic field will no loner be divergence free (in general). Therefore, to explicitly enforce the divergence free conditions on the data assimilation variables without making the systems overdetermined, we also introduce a potential field, .
In the original system, as well as in the following systems, we work with periodic boundary conditions, and assume that and q have zero spatial average. First, we have the following algorithm which utilizes measurements taken on all components (so measuring and ):
subject to the initial conditions .
Next, using measurements only on the first components of and (which is equivalent to measuring and ):
subject to the initial conditions .
Finally, only taking measurements on (which would in practice still require recording measurements on both and ):
subject to the initial conditions .
Although we chose to consider taking measurements on the first components of and in System 2.4, we could instead use the second components with no substantial differences. Likewise, in System 2.5 we could also consider taking measurements on and we would obtain similar results.
In the above we chose to make the initial conditions 0, but in fact the initial conditions may be chosen essentially arbitrarily, albeit in accordance with the existence theorems. Theorem 3.8 additionally requires that the initial conditions satisfy an upper bound of the form (16).
Here we first constructed the Elsässer variables from the original variables and after nondimensionalizing, and then proceeded to define the various data assimilation algorithms and variables. However, since the transformations were linear, if we were to define each data assimilation algorithm using the original variables, in the process defining data assimilation variables and , and then nondimensionalize and change to the Elsässer variables, we would arrive at the same systems above. So, all our results apply to the corresponding algorithms formulated in terms of the original variables.
Note also that although the results are framed in terms of the Elsässer variables, by the triangle inequality convergence of to and to implies convergence of and to and respectively.
We define weak solutions for all the systems mentioned in the distributional sense in the usual way. See [57] for a precise definition in the case of (3) (the other systems are similar). In addition to being a weak solution, we say (or ) is a global strong solution of (11) (or (12), (13), or (14)) if
In [57], it was shown that if and , then there exists a unique global strong solution to (3) (which can be transformed to a solution of (11)). Therefore, we will be assuming that, in addition to being space periodic and divergence free,
The proofs of the corresponding existence and uniqueness results for Systems 2.3–2.5 are similar, and are omitted. We only state and prove the corresponding convergence results.
Before we get to the main theorems, we first state the following bounds for the reference solution to the MHD system. Moreover, we prove (15), which follows standard arguments from the Navier–Stokes theory (see, e.g., [17,60]). The proofs of (16) and (17) can be obtained by modifying the corresponding proofs from the Navier–Stokes theory in a similar way (see, e.g. [20,57] for more details on (16) and the appendix of [23] for (17)).
(Upper Bounds on Solutions of the MHD).
Letbe a solution of (
11
). Then there is aand constantsandsuch that for alland any,
See the appendix. □
Statements of the results
Results for type 1 interpolants
Letbe a strong solution of (
11
) which at timehas evolved enough so that Proposition
2.9
holds with. Letsatisfy (
1
), where(so). Then there is a unique strong solution,, of (
12
) corresponding towhich exists globally in time, and furthermoreexponentially as.
Letbe a strong solution of (
11
) which at timehas evolved enough so that Proposition
2.9
holds with. Letsatisfy (
1
), where(so). Then there is a unique strong solution,, of (
13
) corresponding towhich exists globally in time, and furthermoreexponentially as.
Letbe a strong solution of (
11
) which at timehas evolved enough so that Proposition
2.9
holds with. Letsatisfy (
1
), where(so). Then there is a unique strong solution,, of (
14
) corresponding towhich exists globally in time, and furthermoreexponentially as.
In the next three theorems, by using the convergence results we just established, we show that solutions of (12), (13), and (14) will converge exponentially in time to the reference solution in the stronger topology of the -norm.
Letbe a strong solution of (
11
) which at timehas evolved enough so that Proposition
2.9
holds with. Letsatisfy (
1
), where(so). Then there is a unique strong solution,, of (
12
) corresponding towhich exists globally in time, and furthermoreexponentially as.
Letbe a strong solution of (
11
) which at timehas evolved enough so that Proposition
2.9
holds with. Letsatisfy (
1
), where(so). Then there is a unique strong solution,, of (
13
) corresponding towhich exists globally in time, and furthermoreexponentially as.
Letbe a strong solution of (
11
) which at timehas evolved enough so that Proposition
2.9
holds with. Letsatisfy (
1
), where(so). Then there is a unique strong solution,, of (
14
) corresponding towhich exists globally in time, and furthermoreexponentially as.
Observing the Poincaré inequality, the results of Theorems 3.4–3.6 seem to imply those of Theorems 3.1–3.3, but the spatial resolution is required to be slightly finer for the results. Also, based on our analysis, there may be a longer period of time that must pass before exponential convergence is observed in the -norm than in the -norm (see the estimates in (44) and (47)). However, we point out that in computational results regarding data assimilation in the context of the one-dimensional Kuramoto–Sivasinsky equation, convergence times for both norms are almost identical (cf. [47] for more details).
Results for type 2 interpolants
Letbe a strong solution of (
11
), which at timehas evolved enough so that Proposition
2.9
holds with. Thenandcan be chosen so that ifsatisfies (
2
) then there is a unique strong solutionof (
13
) corresponding towhich exists globally in time, andexponentially as.
Similar theorems hold for the cases of measurements on all variables and one Elsässer variable (although not as direct corollaries, since the dynamical systems involved are slightly different). However, in the case of measuring all variables we do not find much improvement in the restrictions on h and μ.
Determining interpolants
In order to prove that there are finitely many (say N) determining modes, one needs to show that if and are different solutions of (11) with possibly different forcing terms and initial data, then knowledge that is sufficient to conclude that , where denotes the projection onto the modes with magnitude at most N. In general, we replace by a different operator, say , and ask the question of whether the knowledge inherent in is “determining”.
In the following theorems, we show that the data assimilation results we have obtained can be adapted to show that the interpolant operators, , are determining. We do this by first generalizing the convergence results we developed in the previous theorems to allow for the evolution equations of the reference solution and the data assimilation solution to have different forcing terms, which converge in as , at the cost of losing the exponential rate of convergence of the solutions. We also allow for the reference solution to be perturbed by a function which decays in .
We illustrate the ideas for the algorithm studied in Theorem 3.1, i.e. with measurements taken on all variables and for satisfying (1), but the results can be obtained for all the other cases as well. So, we can show that operators which satisfy (1) or (2) and use measurements on or , are determining in the sense of convergence in and .
Letsatisfy (
1
) and letbe a reference solution of (
11
). Then if μ and h satisfy the hypotheses of Theorem
3.1
, and ifandas, there are uniqueandwhich satisfy the following modified version of (
12
):
subject to the initial conditions ,
and furthermore,as.
In the next theorem we illustrate the result that if an interpolant satisfies the conditions for the generalized data assimilation theorem then is determining, for the case of the generalized version of Theorem 3.1. Note that the projection onto the low modes, , is an example of an interpolant operator for which the theorem applies, provided that . Hence, the following theorem shows that there are finitely many determining modes for instance.
Letandbe solutions of (
11
) with forcing termsandrespectively, and suppose that.
Proofs of convergence results with type 1 interpolants
Before we get to the proofs of the main theorems, we first collect the various estimates needed for the bilinear term in the following lemma.
Letbe divergence free. Then the following inequalities hold for anyor
See the appendix. □
The following lemma will be used in our analyses of the algorithms using measurements on only the first components of the reference solutions, where we will need to make use of (8), (9), or (21). The proof is elementary, and therefore omitted.
Letfor someThen,
Let and
Then satisfies:
Using the fact that we write:
Taking the inner product with we obtain:
Now, by the divergence free condition,
By applying Cauchy–Schwarz inequality and (4),
and by rewriting we have:
Thus, we obtain:
where in the last three lines we used Cauchy–Schwarz inequality, the definition of , and Young’s inequality. This leaves us with:
Proceeding the same way for , we have the following equations:
We estimate the integrals in these equations using (19), with and , and obtain
Then, adding (24) and (25), we obtain
Thus, defining and , we have
where , provided that .
By Proposition 2.9 with , ψ satisfies (10b) and if
then ψ also satisfies (10a), so we can apply Proposition 2.1 to Y and conclude that converges exponentially in time to .
The requirement on h is
so . □
Let and Then satisfies:
Using the fact that we write:
Taking the inner product with we obtain:
Now, by the divergence free condition, we have:
By applying Cauchy–Schwarz inequality and (4),
and by rewriting we have:
Thus, we obtain:
where in the last three lines we used the Cauchy–Schwarz inequality, the definition of , and Young’s inequality. This leaves us with:
or equivalently,
Now we apply Lemma 4.1 to estimate the nonlinear term with (21), yielding:
Proceeding similarly with we obtain:
Now, adding (29) and (30) and defining ,
Since ,
provided that and by choosing
We want to apply Lemma 4.2 to the logarithmic terms in (31). To this end note that by (6), , so Next, we write
and consider
By Lemma 4.2,
Hence, using (32) and defining , we rewrite (31) as
By (6),
and so
Let
and in order to apply Proposition 2.1 we only need to show that ψ satisfies (10a) and (10b). It is sufficient to show that for some ,
and
In fact, (35) follows directly from (16) with the given there.
To see (34), by Proposition 2.9 with , we have:
Therefore, (34) holds by choosing . In addition, the requirement implies . □
Let and Similarly to how we showed (28), the equation we obtain for is
but now the equation for is
We estimate the integral in (36) using (19), so (36) becomes:
Similarly, we estimate the integral in (37) using (20), and get:
Adding (38) and (39),
Now, if we choose
and , then .
Also, by choosing , we have
Then by applying (6) we obtain Hence, defining and , we have:
where Using Proposition 2.9 similarly as before, with , ψ satisfies (10b) as well as (10a) provided that
and
By choosing such a μ and δ, we can apply Proposition 2.1 to conclude that converges exponentially in time to .
Now the requirement we needed on h implies
so □
Proof of convergence results with type 1 interpolants
By denoting and and subtracting the equations for and , we obtain the following equations for and
Taking the inner product with and , respectively, we obtain:
Then, by the divergence-free condition,
and similarly
Also, by applying Cauchy–Schwarz inequality and (4), we have
Rewriting we have,
and similarly,
Adding up the equations for and , we obtain
Due to the properties of , we have
and similarly, we obtain
Next, we estimate the nonlinear terms. First, by Hölder’s and Sobolev inequalities, we obtain
where we used Poincaré’s and Young’s inequalities. The estimate for is similarly, i.e., we have
Regarding , we first rewrite it as
In order to estimate I, we first observe that by the periodic boundary conditions, we have
Thus, we integrate by parts and proceed to estimate I as
By similar estimates and the analogy of (41) for , i.e.,
we estimate as
By a similar approach, we have
and is bounded by
while we estimate as
Combining all the above estimates, we obtain
Now choose h such that
Thus, we have
Moreover, by Theorem 3.1, we know that after a sufficiently large time , and are small enough. so that we have
which implies that , so we have:
Define , and by appealing to Proposition 2.9, we see that is bounded by some number . Also, by Theorem 3.1 we know that there exists constants such that . Putting all of this together, we have the following for all :
Therefore, exponentially as as long as μ and h satisfy the conditions of Theorem 3.1, as well as the new requirement (43). So, choosing
we have exponential convergence. □
Next, we prove the decay estimates for the data assimilation scenario where measurement is only on and .
We still denote the difference of solutions to (11) and (13) by and . Similarly to the beginning of the proof of Theorem 3.4, we have
as well as
and
The estimates for the nonlinear terms are also similar. Namely, we have
and
Also, by rewriting
we obtain
and
Estimates for
also follow similarly, and we obtain
and
Combining all the above estimates, we obtain
We choose h such that
In view of Theorem 3.2, after sufficiently large time , and are small enough so that
Thus, . Let us denote . Then, for all , by applying Poincaré’s inequality to the second term on the left-hand side of (45), it follows, due to (46), that
where and chosen so that is such that for all (this is permitted due to Theorem 3.2). This implies
Integrating, we arrive at
(Note that, if necessary, one may choose slightly smaller so that .) In particular, decays exponentially in time for all , with h and μ chosen so that
and
Thus, the proof of Theorem 3.5 is complete. □
The proof goes similarly as that of Theorem 3.5. For the sake of simplicity, we omit the details here. □
Proofs of the results for type 2 interpolants
Letbe divergence free. Then the following inequalities hold:
See the appendix. □
In the following proof of Theorem 3.8, we simultaneously establish a bound like (16) for the data assimilation solution, because the proof requires such an estimate.
Since is a strong solution and , there is a largest time such that
Suppose that .
Then we know that
Let and Then we have the following equation for :
Taking the inner product with , we obtain:
Now, by the divergence free condition, we have:
and by applying Cauchy–Schwarz inequality and (4),
Rewriting we have,
so we obtain:
By the properties of , we have
Therefore,
Note that and as long as
Now we estimate the nonlinear terms using Lemma 4.3. By (48), we obtain
so by applying (4), we obtain
Also, we use (6) to write .
For the other term, we first apply (49), and obtain
Then, by (4), we have
Combining these estimates with (52), we have:
where
Adding (54) with the corresponding inequality for , we obtain:
Next, we write
and
Then, by defining
and
by (6) we can rewrite (55) as:
Now we apply Lemma 4.2 and conclude that
Using (6) again, we have
so by defining
and
we obtain:
Thus, as long as we choose , we conclude by Gronwall’s inequality that
By (50), (16), and (17),
so on the time interval , such a μ is available. Specifically, it is sufficient to choose
where , so
Therefore, for all , we obtain
This implies that, in fact,
which is a contradiction to (50).
Hence we have and converges exponentially in time to in the norm, and we have established the estimate:
Also, our restriction on μ (58) is in fact sufficient to guarantee convergence on , with our restriction (53) on h, which we see now means we can choose
□
Determining interpolants
The proof proceeds exactly as that of Theorem 3.1, where , with a few differences. As before, we let and then we obtain a differential inequality for . We get the same inequality as before but with two extra terms.
After subtracting the equations for and , we have for the forcing term, and after taking the inner product with we have
Also, we have , and after taking the inner product with , we obtain
We have similar additions for the inequality we derive for .
Thus, letting and proceeding as before, we eventually get:
where
and
Since and , we have . Therefore, by Proposition 2.1, as . □
Let . Then h, , and μ satisfy Theorem 3.1 with as the reference solution.
Let be the corresponding solution.
Then and , and for some , q, and satisfy the following equations:
Therefore, setting and , and and , we see that must be the unique solution guaranteed by Theorem 3.10, with as the reference solution.
Therefore and .
Thus,
and
□
Concluding remarks
We have shown that, in the language of the reformulated equations, solutions of the data assimilation equations will converge to the corresponding true values in , even if measurements are only taken for only one of and . This equates to having to take measurements on either or . Could one prove that it is sufficient to collect data on just or just and still get convergence, similar to the result for the reformulated variables?
If one were to consider collecting data only on the magnetic field, , then the problem is evident when we take for all , because we then have and satisfying the Navier–Stokes equations with different initial conditions and no data assimilation. Hence, there is an asymmetry between the original system and the reformulated system.
The answer to the question for collecting data on the velocity field, , is open. However, since we’ve demonstrated that the algorithm works with knowledge of only the sum of measurements on and , it may be that the knowledge of the velocity field is what makes this work, and so a -measurement only algorithm is hopeful. However, since it seems we shouldn’t be able to prove the convergence of a -measurement only algorithm, and the Elsässer variable formulation does not distinguish and , a proof of a -measurement only algorithm would have to be in terms of the original variables.
Footnotes
Acknowledgement
The research of Animikh Biswas and Joshua Hudson was partially supported by NSF grant DMS-1517027. The research of Adam Larios was partially supported by NSF grant DMS-1716801.
References
1.
M.M.Adams, D.R.Stone, D.S.Zimmerman and D.P.Lathrop, Liquid sodium models of the Earth’s core, Progress in Earth and Planetary Science2(1) (2015), 1–18. ISSN 2197-4284. doi:10.1186/s40645-015-0032-y.
2.
D.A.Albanez, H.J.Nussenzveig Lopes and E.S.Titi, Continuous data assimilation for the three-dimensional Navier–Stokes-α model, Asymptotic Anal.97(1–2) (2016), 139–164. doi:10.3233/ASY-151351.
3.
M.U.Altaf, E.S.Titi, O.M.Knio, L.Zhao, M.F.McCabe and I.Hoteit, Downscaling the 2D Benard convection equations using continuous data assimilation, 2015, ArXiv e-prints arXiv:1512.04671.
4.
A.Azouani, E.Olson and E.S.Titi, Continuous data assimilation using general interpolant observables, J. Nonlinear Sci.24(2) (2014), 277–304. ISSN 0938-8974. doi:10.1007/s00332-013-9189-y.
5.
A.Azouani and E.S.Titi, Feedback control of nonlinear dissipative systems by finite determining parameters – a reaction-diffusion paradigm, Evol. Equ. Control Theory3(4) (2014), 579–594., ISSN 2163-2480. doi:10.3934/eect.2014.3.579.
6.
H.Bessaih, E.Olson and E.S.Titi, Continuous data assimilation with stochastically noisy data, Nonlinearity28(3) (2015), 729–753. ISSN 0951-7715. doi:10.1088/0951-7715/28/3/729.
7.
A.Biswas and V.R.Martinez, Higher-order synchronization for a data assimilation algorithm for the 2D Navier–Stokes equations, Nonlinear Anal. Real World Appl.35 (2017), 132–157. ISSN 1468-1218. doi:10.1016/j.nonrwa.2016.10.005.
8.
J.Bloxham, D.Gubbins and A.Jackson, Geomagnetic secular variation, Phil. Trans. R. Soc. A329(1606) (1989), 415–502. doi:10.1098/rsta.1989.0087.
9.
Z.Bradshaw and Z.Grujić, Energy cascades in physical scales of 3D incompressible magnetohydrodynamic turbulence, J. Math. Phys.54(9) (2013), 093503. ISSN 0022-2488. doi:10.1063/1.4819295.
10.
Z.Bradshaw and Z.Grujić, On the transport and concentration of enstrophy in 3D magnetohydrodynamic turbulence, Nonlinearity26(8) (2013), 2373–2390. ISSN 0951-7715. doi:10.1088/0951-7715/26/8/2373.
11.
H.-P.Bunge, M.A.Richards and J.R.Baumgardner, Mantle-circulation models with sequential data assimilation: Inferring present-day mantle structure from plate-motion histories, Phil. Trans. R. Soc. A360(1800) (2002), 2545–2567. doi:10.1098/rsta.2002.1080.
12.
C.Cao, I.G.Kevrekidis and E.S.Titi, Numerical criterion for the stabilization of steady states of the Navier–Stokes equations, Indiana Univ. Math. J.50(Special Issue) (2001), 37–96, Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000). ISSN 0022-2518. doi:10.1512/iumj.2001.50.2154.
13.
J.Chandrasekar, A.J.Ridley and D.S.Bernstein, A comparison of the extended and unscented Kalman filters for discrete-time systems with nondifferentiable dynamics, Proc. of the 2007 American Control Conference2 (2007), 4431–4436. doi:10.1109/ACC.2007.4283100.
14.
S.Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics, Clarendon Press, Oxford, 1961.
15.
B.Cockburn, D.A.Jones and E.S.Titi, Determining degrees of freedom for nonlinear dissipative equations, C. R. Acad. Sci. Paris Sér. I Math.321(5) (1995), 563–568. ISSN 0764-4442.
16.
C.G.Constable, C.L.Johnson and S.P.Lund, Global geomagnetic field models for the past 3000 years: Transient or permanent flux lobes?, Phil. Trans. R. Soc. A358(1768) (2000), 991–1008. doi:10.1098/rsta.2000.0570.
17.
P.Constantin and C.Foias, Navier–Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988, p. 190. ISBN 0-226-11548-8; 0-226-11549-6.
18.
R.Daley, Atmospheric Data Analysis, Cambridge Atmospheric and Space Science Series, Cambridge University Press, 1993. ISBN 9780521458252.
19.
P.A.Davidson, An Introduction to Magnetohydrodynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001, p. 431. ISBN 0-521-79487-0. doi:10.1017/CBO9780511626333.
20.
G.Duvaut and J.-L.Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal.46 (1972), 241–279. ISSN 0003-9527.
21.
W.M.Elsasser, The hydromagnetic equations, Phys. Rev.79 (1950), 183. doi:10.1103/PhysRev.79.183.
22.
A.Farhat, M.S.Jolly and E.S.Titi, Continuous data assimilation for the 2D Bénard convection through velocity measurements alone, Phys. D303 (2015), 59–66. ISSN 0167-2789. doi:10.1016/j.physd.2015.03.011.
23.
A.Farhat, E.Lunasin and E.S.Titi, Abridged continuous data assimilation for the 2D Navier–Stokes equations utilizing measurements of only one component of the velocity field, J. Math. Fluid Mech.18(1) (2016), 1–23. doi:10.1007/s00021-015-0225-6.
24.
A.Farhat, E.Lunasin and E.S.Titi, Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements, Journal of Mathematical Analysis and Applications438(1) (2016), 492–506. doi:10.1016/j.jmaa.2016.01.072.
25.
A.Farhat, E.Lunasin and E.S.Titi, Continuous data assimilation for a 2D Bénard convection system through horizontal velocity measurements alone, Journal of Nonlinear Science27(3) (2017), 1065–1087. ISSN 1432-1467.
26.
A.Farhat, E.Lunasin and E.S.Titi, On the Charney conjecture of data assimilation employing temperature measurements alone: The paradigm of 3D planetary geostrophic model, 2016, arXiv:1608.04770.
27.
C.Foias, O.Manley, R.Rosa and R.Temam, Navier–Stokes Equations and Turbulence, Encyclopedia of Mathematics and Its Applications, Vol. 83, Cambridge University Press, Cambridge, 2001, p. 347. ISBN 0-521-36032-3. doi:10.1017/CBO9780511546754.
28.
C.Foias, O.Manley, R.Temam and Y.M.Trève, Asymptotic analysis of the Navier–Stokes equations, Physica D: Nonlinear Phenomena9(1–2) (1983), 157–188. doi:10.1016/0167-2789(83)90297-X.
29.
C.Foias, C.F.Mondaini and E.S.Titi, A discrete data assimilation scheme for the solutions of the two-dimensional Navier–Stokes equations and their statistics, SIAM J. Appl. Dyn. Syst.15(4) (2016), 2109–2142. ISSN 1536-0040. doi:10.1137/16M1076526.
30.
C.Foia̧s and G.Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier–Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova39 (1967), 1–34. ISSN 0041-8994. http://www.numdam.org/item?id=RSMUP_1967__39__1_0.
31.
C.Foias and R.Temam, Determination of the solutions of the Navier–Stokes equations by a set of nodal values, Math. Comp.43(167) (1984), 117–133. ISSN 0025-5718. doi:10.1090/S0025-5718-1984-0744927-9.
32.
C.Foias and E.S.Titi, Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity4(1) (1991), 135–153. ISSN 0951-7715. doi:10.1088/0951-7715/4/1/009.
33.
A.Fournier, C.Eymin and T.Alboussière, A case for variational geomagnetic data assimilation: Insights from a one-dimensional, nonlinear, and sparsely observed MHD system, Nonlinear Proc. Geoph.14 (2007), 163–180. doi:10.5194/npg-14-163-2007.
34.
A.Fournier, G.Hulot, D.Jault, W.Kuang, A.Tangborn, N.Gillet, E.Canet, J.Aubert and F.Lhuillier, An introduction to data assimilation and predictability in geomagnetism, Space Sci. Rev.155(1) (2010), 247–291. ISSN 1572-9672. doi:10.1007/s11214-010-9669-4.
35.
K.Foyash, M.S.Dzholli, R.Kravchenko and È.S.Titi, A unified approach to the construction of defining forms for a two-dimensional system of Navier–Stokes equations: The case of general interpolating operators, Uspekhi Mat. Nauk69(2(416)) (2014), 177–200. ISSN 0042-1316. doi:10.4213/rm9583.
36.
M.Gesho, E.Olson and E.S.Titi, A computational study of a data assimilation algorithm for the two-dimensional Navier–Stokes equations, Commun. Comput. Phys.19(4) (2016), 1094–1110. ISSN 1815-2406.
37.
N.Glatt-Holtz, I.Kukavica, V.Vicol and M.Ziane, Existence and regularity of invariant measures for the three dimensional stochastic primitive equations, J. Math. Phys.55(5) (2014), 051504. ISSN 0022-2488. doi:10.1063/1.4875104.
38.
C.Groth, D.L.De Zeeuw, T.I.Gombosi and K.G.Powell, Global three-dimensional MHD simulation of a space weather event: CME formation, interplanetary propagation, and interaction with the magnetosphere, J. Geophys. Res.: Space Phys.105(A11) (2000), 25053–25078. doi:10.1029/2000JA900093.
39.
K.Hayden, E.Olson and E.S.Titi, Discrete data assimilation in the Lorenz and 2D Navier–Stokes equations, Phys. D240(18) (2011), 1416–1425. ISSN 0167-2789. doi:10.1016/j.physd.2011.04.021.
40.
M.S.Jolly, V.R.Martinez and E.S.Titi, A data assimilation algorithm for the subcritical surface quasi-geostrophic equation, Adv. Nonlinear Stud.17(1) (2017), 167–192. ISSN 1536-1365. doi:10.1515/ans-2016-6019.
41.
M.S.Jolly, T.Sadigov and E.S.Titi, A determining form for the damped driven nonlinear Schrödinger equation – Fourier modes case, J. Differential Equations258(8) (2015), 2711–2744. ISSN 0022-0396. doi:10.1016/j.jde.2014.12.023.
42.
D.A.Jones and E.S.Titi, Determining finite volume elements for the 2D Navier–Stokes equations, Phys. D60(1–4) (1992), 165–174, Experimental mathematics: computational issues in nonlinear science (Los Alamos, NM, 1991). ISSN 0167-2789. doi:10.1016/0167-2789(92)90233-D.
43.
D.A.Jones and E.S.Titi, On the number of determining nodes for the 2D Navier–Stokes equations, J. Math. Anal. Appl.168(1) (1992), 72–88. ISSN 0022-247X. doi:10.1016/0022-247X(92)90190-O.
44.
D.A.Jones and E.S.Titi, Upper bounds on the number of determining modes, nodes, and volume elements for the Navier–Stokes equations, Indiana Univ. Math. J.42(3) (1993), 875–887. ISSN 0022-2518. doi:10.1512/iumj.1993.42.42039.
45.
E.Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, 2003. ISBN 9780521796293.
46.
A.Larios, E.Lunasin and E.S.Titi, Global well-posedness for the Boussinesq–Voigt equations, preprint, arXiv:1010.5024.
47.
A.Larios and Y.Pei, Nonlinear continuous data assimilation, 2017, arXiv:1703.03546.
48.
K.Law, A.Stuart and K.Zygalakis, A Mathematical Introduction to Data Assimilation, Texts in Applied Mathematics, Vol. 62, Springer, Cham, 2015, p. 242. ISBN 978-3-319-20324-9; 978-3-319-20325-6.
49.
K.Li, A.Jackson and P.W.Livermore, Variational data assimilation for a forced, inertia-free magnetohydrodynamic dynamo model, Geophys. J. Int.199(3) (2014), 1662–1676. doi:10.1093/gji/ggu260.
50.
E.Lunasin and E.S.Titi, Finite determining parameters feedback control for distributed nonlinear dissipative systems – a computational study, 2015, arXiv:1506.03709 [math.AP].
51.
P.A.Markowich, E.S.Titi and S.Trabelsi, Continuous data assimilation for the three-dimensional Brinkman–Forchheimer-extended Darcy model, Nonlinearity29(4) (2016), 1292. doi:10.1088/0951-7715/29/4/1292.
52.
O.B.Mendoza, B.De Moor and D.S.Bernstein, Data assimilation for magnetohydrodynamics systems, J. Comput. Appl. Math.189(1–2) (2006), 242–259. ISSN 0377-0427. doi:10.1016/j.cam.2005.03.030.
53.
C.F.Mondaini and E.S.Titi, Postprocessing Galerkin method applied to a data assimilation algorithm: A uniform in time error estimate, 2016, arXiv:1612.06998.
54.
E.Olson and E.S.Titi, Determining modes for continuous data assimilation in 2D turbulence, J. Statist. Phys.113(5–6) (2003), 799–840, Progress in statistical hydrodynamics (Santa Fe, NM, 2002). ISSN 0022-4715. doi:10.1023/A:1027312703252.
55.
T.J.Sabaka, N.Olsen and R.A.Langel, A comprehensive model of the quiet-time, near-Earth magnetic field: Phase 3, Geophys. J. Int.151(1) (2002), 32–68. doi:10.1046/j.1365-246X.2002.01774.x.
56.
R.B.Scott, D.E.McKenzie and D.W.Longcope, Inferring the magnetohydrodynamic structure of solar flare supra-arcade plasmas from a data-assimilated field transport model, Astrophys. J.819(1) (2016), 56. http://stacks.iop.org/0004-637X/819/i=1/a=56. doi:10.3847/0004-637X/819/1/56.
57.
M.Sermange and R.Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math.36(5) (1983), 635–664. ISSN 0010-3640. doi:10.1002/cpa.3160360506.
58.
Z.Sun, A.Tangborn and W.Kuang, Data assimilation in a sparsely observed one-dimensional modeled MHD system, Nonlinear Proc. Geoph.14(2) (2007), 181–192. doi:10.5194/npg-14-181-2007.
59.
B.O.Teixeira, A.Ridley, L.A.Tôrres, L.A.Aguirre and D.S.Bernstein, Data assimilation for magnetohydrodynamics with a zero-divergence constraint on the magnetic field, in: 2008 American Control Conference, IEEE, 2008, pp. 2534–2539.
60.
R.Temam, Navier–Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001, p. 408, Theory and numerical analysis, Reprint of the 1984 edition. ISBN 0-8218-2737-5.
61.
E.S.Titi, On a criterion for locating stable stationary solutions to the Navier–Stokes equations, Nonlinear Analysis: Theory, Methods & Applications11(9) (1987), 1085–1102. doi:10.1016/0362-546X(87)90086-1.
