Spectral Methods Based on the Least Dissipative Modes for Wall...

Theoretical and Computational Fluid Dynamics manuscript No.(will be inserted by the editor)

V. Dymkou · A. Potherat

Spectral Methods Based on the Least DissipativeModes for Wall Bounded MHD Flows

Received: date / Accepted: date

Abstract We present a new approach for the Spectral Direct Numerical Simulation (DNS) of Low-Rm wall bounded magnetohydrodynamic (MHD) flows. The novelty is that instead of using bases likethe usual Tchebychev polynomials, that are easy to implement but incur heavy computational costsin order to resolve the Hartmann boundary layers that arise along the walls, we use a basis made ofelements that already incorporate flow structures such as anisotropic vortices and Hartmann layers.We show that such a basis can be obtained from the eigenvalue problem of the linear part of thegoverning equations with the problem’s boundary conditions. Since this basis is not always orthogonal,we develop a spectral method for non-orthogonal bases. We then demonstrate the efficiency of thismethod on the simple case of a laminar channel flow with periodic forcing. In particular, we show thatthis method eliminates the computational costs incurred by the Hartmann layer, and this for arbitraryhigh magnetic fields B. We then discuss the application of our method to nonlinear, turbulent flowsfor which the number of modes required to resolve the flow completely decreases strongly when Bincreases instead of increasing, as in current Tchebychev-based methods.

Keywords MHD · turbulence · spectral methods · canonical system · channel flow

PACS First · Second · More

1 Introduction

We are interested in obtaining numerical solutions for liquid metal magnetohydrodynamic (MHD)flows in conditions that are realistic enough to be relevant to experiments or applications, and, inparticular, that of the liquid metal blankets used as cooling devices in the nuclear fusion reactor ITER.Realistic conditions in such a case typically involve a duct with a constant magnetic field B of around10T. Under such conditions, the main effect of the Lorentz force is to oppose variations of physicalquantities along the magnetic field lines. On the top of developing a strong anisotropy, this also resultsin very thin boundary layers, the Hartmann layers developing along the walls that intercept the fieldlines (see for instance [35,28,7]). Our purpose is to show how spectral methods can be developed thattake advantage of these specific properties of MHD flows in order to offer an efficient way of solvingcomplex PDEs such as those that govern this problem.

The principle of Spectral Methods consists in looking for a solution for the physical variables (herethe space x and time t dependent velocity field u(x, t) and pressure p(x, t)) under the form of a finite

The authors would like to express their gratitude to the Deutsche ForschunGemeinschaft for its support undergrant PO1210/1-1

Coventry University, Applied Mathematics Research Centre, Tel.: +44 (0) 2476 88 88 65, Fax: +44 (0) 247688 80 80. E-mail: [email protected]

2 V. Dymkou, A. Potherat

expansion over a basis of functions (en)1≤n≤N that spans the space of solutions of the PDE and carriesall of the spatial dependence:

u(x, t) =∑

1≤n≤N

cn(t)en(x). (1)

Once inserted in the governing equations, the problem reduces to a much simpler set of N ODEs forthe coefficients cn(t). In most fluid dynamics problems, the basis (en)1≤n≤N is chosen so that it isorthogonal and that each element satisfies the problem’s boundary conditions (an extensive accountof spectral methods and their application to fluid mechanics can be found in [5,4]). In practice, such abasis is hard to find for problems other than rectangular domains with periodic boundary conditionsor non-slip, impermeable walls. In the first case, Fourier modes can be used that are both easy toimplement and involve low computational costs thanks to the use of Fast Fourier Transforms to dealwith the non-linear terms. When walls are present, orthogonal bases using Tchebychev or Legendrepolynomials are often chosen as Fast Transformation techniques also exist for them. For these reasons,Spectral Methods in Computational Fluid Dynamics (CFD) have been mostly confined to rather sim-ple if not simplistic geometries and been reproached with not being applicable to real cases.The picture gets worse when considering the computational cost. The latter is controlled by the num-ber of modes N required to resolve the flow. In the case of a non-MHD Turbulent flow in a periodicbox, spectral methods don’t bring any additional cost to the estimate of N ∼ Re9/4 given by theKolmogorov theory [23] that reflects the size of the smallest scale present in the flow (Re is a Reynoldsnumber). When walls are involved, however, boundary layers develop along them and need to be finelyresolved. This is done by increasing the order of the polynomials that constitute (en)1≤n≤N , andthereby N and the computational cost. This problem is critical in liquid metal magnetohydrodynamicsas in the simple channel flow problem, the thickness of the Hartmann boundary layers varies as B−1.Typically, for a 10 cm wide channel of gallium in a 10T field, it becomes of only a few µm, thus lettingN increase linearly with B and reach orders of magnitude of 1010, way beyond the reach of current(and foreseeable future) computers. For this reason, recent computations of MHD channel flows, suchas those from [25,3] are limited to moderate magnetic fields and flow intensities.In our recent work we have derived rigorous upper bounds for the attractor dimension for the Hartmannflow problem (channel flow with uniform magnetic field orthogonal to the channel) that proves thatthe number of modes required to resolve the flow varied as N ∼ 1/Ha, where Hartmann number Hacan be thought of as a non dimensional measure of B. The main point here, is that N should actuallystrongly decrease when B increases, and not increase as it does when using Tchebychev polynomialsfor (en)1≤n≤N . We believe that this apparent contradiction simply shows that Tchebychev polynomialsare an artificially costly basis for this problem and that the cost of such calculations could be broughtdown by using a set of function whose element ”mimic” the flow more closely. We have previouslyshown that such a set could be built from the eigenvectors of the linear part of the governing equationsfor the simple case of a periodic domain [30] and that in the case of a channel flow [31], the eigenvectorsalso carried most of the known flow features (although we haven’t calculated them fully explicitly).In the first case, the eigenfunctions obtained were essentially the usual basis of Fourier modes, withthe important difference that they were ordered by growing rate of linear decay. This way, the set ofmodes used to resolve the flow was not isotropically spread in the Fourier space. Instead, they werelocated outside the so-called Joule Cone, precisely where energy-carrying modes had previously beenfound both heuristically and experimentally [26,2]. Using this basis, we were able to derive an MHDequivalent to the Kolmogorov laws and prove that the flow could be fully resolved using significantlyless modes than the isotropic set of modes from the usual Fourier basis [32].In the case of a channel flow, we have found that eigenfunctions of the linear part of the Navier-Stokesequation carried the same anisotropic properties as in the 3D periodic case and that they additionallyfollowed a Hartmann boundary layer profile in the vicinity of the walls. Clearly, only a small numberof these modes would be required to resolve this layer. It is reasonable to expect that such gain can beobtained from any configuration that includes these boundary layers, including duct flows. In general,however, the set of eigenfunctions from the linear part of the equations is not orthogonal, unlike thebasis of Tchebyshev polynomials, so the usual spectral methods need to be adapted.It is precisely at this point that the present work aims at making a step forward: using recent resultsfrom the spectral theory, we establish a method to extract a biorthogonal set of eigenvectors from thelinear part of the Navier-Stokes equation and the related adjoint operator, and use this set in order to

Spectral Methods Based on the Least Dissipative Modes for Wall Bounded MHD Flows 3

reduce the problem to a set of Ordinary Differential Equations. For clarity, we shall then restrict thescope of the present paper to flows in closed domains for which the presentation of the method can besimplified. The extension to open flows poses no major difficulty in principle, but will be left aside hereas the example of closed flows illustrates well enough how the use of a well chosen basis can eliminatethe difficulty posed by the presence of flow structures such as Hartmann boundary layers. Also, we wishto stress here that the method we present potentially applies well beyond the sole scope of liquid metalmagnetohydrodynamics: for any problem where flow structures are not well represented by the basesfrom usual spectral methods, it is worth checking if the spectral analysis of the operators involved in thegoverning equations delivers a set of modes that reasonably mimics the properties of the flow. If this isthe case, building spectral methods based on them may well not only drastically reduce the cost of DNSfor this problem considerably, but also extend the range of flow parameters where spectral DNS canbe used. For this reason too, we will devote a large part of this paper to the presentation of the method.

We shall proceed as follows: in section 2, we recall the governing equations and identify the operatorrelevant to the class of MHD flows we are interested in. In section 3, we first recall some recent math-ematical results that then allow us to build spectral methods that use non-orthogonal bases derivedfrom the spectral analysis of this operator. In section 4, we give a simple example of application ofthis method for the laminar case of a channel flow with a transverse magnetic field, that is relevant tolaboratory experiments on MHD turbulence such as those of [21,19]. We also explain how the methodcan be extended to the fully nonlinear case, and extract a prediction for the first three-dimensionalmode to appear in a quasi-two-dimensional flow from the properties of the least dissipative modes.

2 Governing equations

2.1 Equations for liquid metal magnetohydrodynamics

We shall first recall the equations that govern liquid metal flow magnetohydrodynamics at the scale ofthe laboratory, with the aim of identifying the general form of the operators relevant to our purpose.Boundary conditions will be left unspecified for now, keeping in mind that they are an essential partof the eigenvalue problems that determine the eigenmodes to be used in expansion (1).We consider an electrically conducting, magnetically permeable, incompressible fluid (density ρ, kine-matic viscosity ν and electrical conductivity σ) subject to a uniform, steady magnetic field B = Bez

in a three-dimensional physical domain Ω. Electric currents are generated because of the fluid motionin the magnetic field, that in turn induce a secondary magnetic field. In the frame of the quasi-staticapproximation [35], however, this secondary magnetic field is assumed very small compared to the ex-ternally applied magnetic field B, so the latter can be considered as imposed. This assumption holds inthe limit of small magnetic Reynolds numbers Rm = µ0σUl, where U and l are typical flow velocitiesand lengthscale for the problem considered. In this case, assuming the non-dimensional Ohm’s lawthat relates the electric current density j, and the electric potential φ

j = −∇φ + u × B, (2)

holds (j and φ has been normalised by σBU and σBUl respectively, where U and l are typical velocitiesand lengthscale), the Lorentz force j×B can be expressed linearly as a function of the velocity field u

[35]. Assuming a given external force f , normalised by ‖f‖2/l3/2, (‖ · ‖2 is the usual L2 norm), u andthe pressure p are thus governed by the set of equations:

∂

∂tu(x, t) + (u · ∇)u + ∇p = ∇2u− Ha2∇−2 ∂2u

∂z2+ Gf(x, t),

∇ · u = 0.

(3)

Here the equations have been written under non-dimensional form, normalising velocities by U , lengthsby l, times by l/U and pressure by ρU2. The Grashoff number G = l3/2‖f‖2/ν2 and the Hartmann

number Ha = Bl√

σ/ρν are the governing parameters for the problem. The usual Reynolds number


Re = Ul/ν gives a measure of the intensity of the flow but cannot be a priori related to G and Ha.For (3) to be defined, the boundary conditions on ∂Ω must ensure that the related laplacian operator∇2 is invertible. In many liquid metal MHD problems, periodic boundary conditions or impermeable,no-slip walls are considered. The latter express as homogeneous Dirichlet conditions for u. Since (3)is potentially of 4th order, additional boundary conditions are required, that physically correspond tothe conditions on the electric current at the boundary (they can express, for example, that some partsof ∂Ω are electrically insulating or conducting walls). We have previously shown that these boundaryconditions could be expressed in terms of the velocity using [31]:

j = ∇−2∂z∇× u. (4)

Finally, the flow is assumed known at t = 0, which provides the initial condition:

u(t = 0) = u0. (5)

2.2 General operator form

In order to develop spectral methods that are ”tailored” to the problem we wish to solve, we shall nowwrite (3) and associated boundary conditions under the form of an initial boundary value problem(IBVP) and identify the operator that will yield a sequence (en)1≤n≤N relevant to the problem. Forthis, we first need to identify the functional spaces where the solution of our problem are to be sought:following [13], let Hm(Ω) be the Sobolev space of functions from L2(Ω) whose derivatives of order upto m belong to L2(Ω). (Hm(Ω))3 is the space of three-dimensional vector fields with components fromHm(Ω). By Lp(Ω;B), 1 ≤ p < ∞, we denote the set of functions v defined on Ω ⊂ R

n with images in

the given Banach space B, for which the norm ||v||Lp(Ω;B) =

(

∫

E

||v||pBdE

)1/p

is finite. Then, defining

spaces V, V 0, V 1 and V 2 as:

V = v(x) ∈ (H1(Ω))3 : div v = 0 in Ω,v satisfies the boundary conditions on ∂Ω,

V 0 = the closure of V in (L2(Ω))3,

V 1 = the closure of V in (H1(Ω))3,

V 2 = V 1 ∩ (H2(Ω))3,

(6)

then, assuming f(·) ∈ L2(0, T ; V 0) and u0 ∈ V 1 and using the Helmholtz-Leray decomposition ofvector field u (see [13]), (3) can be proved equivalent to the following Cauchy problem for u:

u(t) = Lu + B(u) + f(t),

u(t)∣

∣

t=0= u0,

(7)

where u(t) = u(•, t) and u(·) ∈ L2(0, T ; V 2) : u(t) ∈ L2(0, T ; V 0), f(t) = f(•, t) and the operators Land B are defined as:

Lu =1

ReP

(

∇2u − Ha2∇−2 ∂2u

∂z2

)

∀u ∈ D(L), (8)

B(u) = P(u · ∇)u ∀u ∈ D(L), (9)

where D(L) = D(B) = V ∩ (H2(Ω))3 and P denotes the Leray Projector:

P : (L2(Ω))3 → V 0. (10)

Finally, the pressure is recovered thanks to the properties of the Helmholtz-Leray projector. Theseensure that it can be expressed as a quadratic function of u that satisfies:

(I − P )

(

(u · ∇)u − 1

Re(∇2u − Ha∇−2 ∂2u

∂z2)

)

= −∇p (11)


and p is obtained by solving:

∇2p = −∇ · (u · ∇u) , (12)

with boundary conditions for p on ∂Ω deduced from that on u, as in [13].At this point, the essential difference between (7) and the operator form of the usual Navier-Stokesequations lies in the expression of L. In the non-MHD case, L is the Stokes operator that representsviscous dissipation and is isotropic if the boundary conditions are. In the case of a three-dimensionalperiodic domain, its eigenfunctions are the orthonormal basis of Fourier modes, spread isotropicallyin Fourier space, that are used in usual Spectral Direct Numerical Simulations. In the MHD case, theoperator L includes the effect of the Lorentz force that introduces anisotropic Joule dissipation, bydamping more strongly modes with their wavevector aligned with the magnetic field [26,6]. We haveshown both in the case of a periodic domain [30] and in that of a channel [31], that the eigenvectors ofL reproduce these physical properties very finely. On these grounds, it is expectable that expansionsof u on this set of eigenmodes would require much less modes than an expansion over a set of modesthat ignores these properties, in order to fully resolve the flow. We shall therefore now adapt the usualspectral techniques used to solve (7) so as to implement this particular sequence of eigenvectors. Sincethe sequence of eigenvectors of L is not orthogonal in the general case, the first step is to establish amethod to perform a spectral discretization of (7) over a non-orthogonal basis.

3 General algorithm for orthogonal and non-orthogonal bases

3.1 The canonical system of L

We shall first recall some results of Spectral Operator Theory that will allow us to construct a basisof functions over which the solution of (3) can be expanded. These results are not new as such asmathematicians have been using them for some years now. To our knowledge, however, they have notyet been applied to the resolution of fluid mechanics problems. Since the work presented here testifiesof their practical relevance to such problem, it is worth underlining the most significant results here.A more in-depth account of these concepts can be found in [17,18,14,15,11] and their application tolinear acoustics in [12].

Let us consider an Operator L acting on a Hilbert space H , with a scalar product denoted (·, ·).When applied to MHD flows, L is associated to the linear part of the Navier-Stokes equations, H spansthe space of possible solenoidal solutions and the scalar product is that associated to the L2 norm.The prime canonical system of L is defined by:

(λI − L)ep,0 = 0,

ep,0 + (λI − L)ep,1 = 0,

· · · · · · · · · ,

ep,m−1 + (λI − L)ep,m = 0,

· · · · · · · · · ,

ep,Mp−1 + (λI − L)ep,Mp= 0.

(13)

λ spans the spectrum of eigenvalues σ(L) of L, (ep,0(λ)) are the usual eigenvectors associated to λ and1 ≤ p ≤ P (λ), where P (λ) is the dimension of the eigenvector space for the eigenvalue λ. When Mp ≥ 1,then (ep,m)m∈1..Mp(λ) are the associated vectors that, together with (ep,0(λ)) form the canonical systemE of L. The maximum possible number m of associated vectors for ep,0 is denoted by Mp = Mp(λ)and (Mp + 1) is called the multiplicity of the eigenvector ep,0. In finite dimensional matrix spaces, thecanonical system essentially gives the Jordan decomposition of a matrix. The adjoint canonical system


E† is obtained from the adjoint operator L† :

(λ∗I − L†)ǫp,0 = 0,

ǫp,0 + (λ∗I − L†)ǫp,1 = 0,

· · · · · · · · · ,

ǫp,m−1 + (λ∗I − L†)ǫp,m = 0,

· · · · · · · · · ,

ǫp,Mp−1+ (λ∗I − L†)ǫp,Mp

= 0,

(14)

where the star superscript stands for complex conjugate. In classical spectral methods, where Fourieror usual polynomials are used, the related operator is self adjoint so eigenvectors form an orthogonalset. In the more general case, L 6= L† so this property gives way to a somewhat weaker one:

Theorem 1 After proper normalisation of ǫp,Mpfor each eigenvalue the introduced canonical systems

E and E† are biorthonormal

(ep,m(λi), ǫl,n(λ∗w)) = δi,wδp,lδm,Mp−n. (15)

In order to expand the solution of (3) or any other vector on this biorthonormal system, we needto define the transformations that yield the coefficients of this expansion:

Definition 1 The Multi-Functional Transformation T is the set of transformations Tp,m(λi), whichare constructed using the canonical system E† of the adjoint spatial operator L† and act on elementsf ∈ H as

Tp,m(λi)f(•) = fp,m(λi) = (f(•), ǫp,m(•, λ∗i )), (16)

where p = 1, . . . , Pi = P (λi), m = 0, . . . , Mp are defined for each λi ∈ σ(L) by the canonical systemand (·, ·) denotes the inner product in the Hilbert space H .

An immediate consequence is that:

Theorem 2

Tp,m(λi)Lf(•) = λifp,m(λi) + fp,m−1(λi), m = 0, . . . , Mp, (17)

where, by convention fp,µ(λi) ≡ 0, µ < 0.

For orthogonal systems such as that of Fourier modes, the adjoint eigenvectors and complex conju-gate eigenvalues become the prime eigenvectors and eigenvalues. In this case, Tp,m(λi) is the usualFourier transform.

Using Tp,m(λi), we can now expand the elements f of H over E. The last step before we canuse an expansion over E in order to solve (3), is to define an inverse transform that will allow us toreconstruct any element f ∈ H from the sequence of coefficients fp,m. For this to be possible, bothE and E† must be bases of H . [16] has proved that it was always the case when the dimension ofthe underlying physical space was 1 (i.e for ODEs). The extension of this result to physical spaces ofdimension 2 or more is still part of ongoing work in the field of Operator Theory, so the fact that Eand E† are bases of H must be proved in each particular case. The proof that it is is however hardlyever simple to establish so for the practical application at sight here, we shall simply assume that Eand E† are bases of H . Then, the inverse transformation is given by the following series

T −1fp,m(λi) = f(•) =∑

λi∈σ(L)

Pi∑

p=1

Mp∑

m=0

fp,m(λi)ep,Mp−m(•, λi). (18)

In practice, should a case arise where the canonical system doesn’t span the whole space of solutions,a basis for the ’missing’ subspace would have to be added to the canonical system.


3.2 Spectral discretization

In order to find a numerical solution of the Navier-Stokes equations under the form of an expansionover the elements of basis E, we shall now adapt the discretization procedure that yields a system ofODEs (see for instance [5]) to non-orthogonal bases. The first step consists of expanding the velocityfield, the non-linear terms and the externally applied force density over E:

u(x, t) =∑

λi∈σ(L)

Pi∑

p=1

Mp∑

m=0

up,m(λi, t)ep,Mp−m(x, λi),

B(u(x, t)) =∑

λi∈σ(L)

Pi∑

p=1

Mp∑

m=0

Bp,m(λi, t)ep,Mp−m(x, λi),

f(x, t) =∑

λi∈σ(L)

Pi∑

p=1

Mp∑

m=0

fp,m(λi, t)ep,Mp−m(x, λi)

(19)

and then inject these expansions into (7):

∑

λi∈σ(L)

Pi∑

p=1

Mp∑

m=0

Dtup,m(λi, t)ep,Mp−m(x, λi) =∑

λi∈σ(L)

Pi∑

p=1

Mp∑

m=0

up,m(λi, t)Lep,Mp−m(x, λi)

+∑

λi∈σ(L)

Pi∑

p=1

Mp∑

m=0


+∑

λi∈σ(L)

Pi∑

p=1

Mp∑

m=0

Bp,m(λi, t)ep,Mp−m(x, λi).

(20)

From the definition of the canonical system (see (13)) we have

∑

λi∈σ(L)

Pi∑

p=1

Mp∑

m=0

Dtup,m(λi, t)ep,Mp−m(x, λi) =∑

λi∈σ(L)

Pi∑

p=1

Mp∑

m=0

up,m(λi, t)ep,Mp−m−1(x, λi)

+∑

λi∈σ(L)

Pi∑

p=1

Mp∑

m=0

up,m(λi, t)λiep,Mp−m(x, λi)

+∑

λi∈σ(L)

Pi∑

p=1

Mp∑

m=0


+∑

λi∈σ(L)

Pi∑

p=1

Mp∑

m=0

Bp,m(λi, t)ep,Mp−m(x, λi).

(21)

Finally, using the property of biorthogonality of the prime and adjoint canonical systems (15), werewrite the problem in the following form

Dtup,m(λi, t) = up,m−1(λi, t) + up,m(λi, t)λi + fp,m(λi, t) + Bp,m(λi, t), (22)

where p = 1, . . . , Pi = P (λi), m = 0, . . . , Mp, λi ∈ σ(L) and up,µ(λi, t) ≡ 0 for µ < 0.

We therefore have:• for m = 0 :

Dtup,0(λi, t) = λiup,0(λi, t) + fp,0(λi, t) + Bp,0(λi, t). (23)


For unsteady flows, multiplying (23) by e−λit, yields:

∂

∂t

(

e−λitup,0(λi, t)

)

= e−λit

(

fp,0(λi, t) + Bp,0(λi, t)

)

. (24)

• and for any m = 1, · · · , Mp:

Dtup,m(λi, t) = up,m−1(λi, t) + λiup,m(λi, t) + fp,m(λi, t) + Bp,m(λi, t). (25)

Again, in the unsteady case, multiplying (24) by e−λit, yields:

∂

∂t

(

e−λitup,m(λi, t)

)

= e−λit

(

up,m−1(λi, t) + fp,m(λi, t) + Bp,m(λi, t)

)

. (26)

This system of ODE can then be solved by means of the usual schemes for the treatment of the timederivatives that reduces this system of ODE to an algebraic one, to be solved at each time step.We shall however not discuss the numerical implementation of these techniques here in detail as our goalis simply to lay the theoretical grounds for spectral methods based on the least dissipative eigenvectorsof L, and illustrate them on a simple example that can be solved without numerical techniques.At this point, it should be pointed out that because the boundary conditions are included in thedefinition of L, the elements of the canonical system satisfy them already. Since discretization definedin this section has been performed without specifying them, (24) and (26) apply virtually to anygeometry. The only requirement for the solution to take the form (19) is that the canonical system ofthe operator L should form a basis.

3.3 General solution for linear problems

For inertialess flows, a general solution can be expressed analytically using the least dissipative modes.Indeed, for such flows, the quadratic term disappears from (3), so the corresponding spectrally dis-cretized problem (26) is reduced to:

for m = 0 Dtup,0(λi, t) = λiup,0(λi, t) + fp,0(λi, t), (27)

for 0 < m ≤ Mp Dtup,m(λi, t) = up,m−1(λi, t) + λiup,m(λi, t) + fp,m(λi, t). (28)

For |λi| < |λmax|, with initial condition:

u0(x) =∑

λi∈σ(L)

Pi∑

p=1

Mp∑

m=0

u0p,m(λi)ep,Mp−m(x, λi). (29)

If the forcing is assumed time-independent, a steady solution U is readily found:

for 0 ≤ m ≤ Mp up,m(λi) =

m∑

k=0

fp,k(λi)

(−λi)m−k+1

, (30)

with transient solution:

for 0 ≤ m ≤ Mp up,m(λi, t) = eλitm

∑

k=0

tm−k

(m − k)!u0p,k

(λi) +

m∑

k=0

fp,k(λi)

(−λi)m−k+1

− eλitm

∑

k=0

k∑

l=0

tk−l

(k − l)!

fp,m−k(λi)

(−λi)l+1.

(31)

For time dependent forcings, the problem still has an exact solution:

for 0 ≤ m ≤ Mp up,m(λi, t) = eλitm

∑

k=0

tm−k

(m − k)!u0p,k

(λi) + eλitIm, (32)

where Im(λi, t) = Im−1(λi, t) +∫ t

0 e−λit′

fp,m(λi, t′)dt′, with Iµ ≡ 0 for µ < 0. The form of (30), (31)

and (32) shows that no mode other than those that constitute the forcing are required to resolve


I

I

Insulating

Walls

Electrically

B

ex

ey

ez

Fig. 1 Geometry of the Channel flow with transverse magnetic field. I: Hartmann boundary layers. In laminar,high Ha regime, the velocity does not vary along ez outside these layers.

the flow completely. In particular, if the flow is forced with one mode only (f = fp,0(λi, t)ep,0), thelaminar solution can be obtained with this only one mode. In the case of MHD flows, this applies toconfigurations involving very thin Hartmann layers, no matter how high the Hartmann number. If, incontrast, such problems were discretized using Tchebychev polynomials to resolve this layer, not onlywould the number of modes required to resolve the Hartmann layer increase drastically with Ha, butthe resolution procedure would require a costly matrix inversion because Tchebychev polynomials donot have the profile of the eigenmodes (and possible associated eigenmodes) of L across the layer. Thesolution would therefore not take such a simple analytical form as that of (30), (31) and (32), andwould have to be obtained numerically.The result presented in this section therefore yields a convenient general solution for linear problems.Once the least dissipative modes are known, the forcing must be projected onto each of them. Thesolution for the velocity field is then obtained by direct application of (32), where the sum only spansthe vectors of the least dissipative modes that are not orthogonal to the forcing.

4 Application to Wall bounded MHD flows

4.1 Problem geometry and boundary conditions

We now consider a channel flow geometry with a homogeneous magnetic field oriented along ez,orthogonal to two parallel, impermeable and electrically insulating walls (or Hartmann walls) locatedat z = −1 and z = 1 respectively (see figure 1). Along these walls, the impermeability and no-slipboundary conditions read:

u|z=−1 = 0, (33)

u|z=1 = 0.

In the x and y directions, the box is assumed periodic, with period 2π:

u|x=−π = u|x=π, (34)

u|y=−π = u|y=π.

In the frame of the Low-Rm approximation the condition that the walls are electrically insulating isexpressed in terms of the velocity using (4) as in [31]:

∇−2∂z∇× u|z=−1 = 0, (35)

∇−2∂z∇× u|z=1 = 0.

Additionally, the constraint imposed by the continuity equation is equivalent to the condition:

∂zu · ez|z=−1 = 0, (36)

∂zu · ez|z=1 = 0.


4.2 Eigenvalue problem

We shall now find the explicit expression of all eigenvectors of L. In our previous work [31], for thegeometry specified in section 4.1, we found the eigenvalues of L in the related V 2 space and showedthat the eigenvectors were of the form:

ep,0 = (A1 exp(iκzz) + A2 exp(−iκzz) + A3 exp(z/δ) + A4 exp(−z/δ)) exp(ik⊥ · r⊥) (37)

= Z(z) exp(ik⊥ · r⊥).

Here, k⊥ = kxex + kyey with (kx, ky) ∈ Z2 and r⊥ = xex + yey. A first striking similarity with the

actual velocity field in channel flows with transverse magnetic fields appears as κz clearly represents awavelength in the ez direction while δ is the thickness of the boundary layer that develops along theHartmann walls. We have also shown in [31] that the formal eigenvalue equation (13) implied that thediscrete sequence of eigenvalues λ was given by the dispersion relation:

λ = −(k2x + k2

y − K) + Ha2 K

k2x + k2

y − K, (38)

where, for a given eigenvalue λ, K can take the values K = −κ2z and K = 1/δ2, so eliminating λ

between (38) with K = −κ2z and (38) with K = 1/δ2, we find that κz and 1/δ are related by:

κ2z

δ2− k2

⊥(k2⊥ − 1

δ2+ κ2

z + Ha2) = 0. (39)

κz, therefore, is no integer. Instead, it is determined by the solvability conditions for the eigenvalueproblem. The latter is solved numerically, as explained below. This, in turn gives us the numericalvalues for δ (39). A detailed analysis of the properties of the sequence of values of δ and κz is presentedin [31], where we show in particular that it incorporates both the laminar and turbulent Hartmannlayer thicknesses. Finally, the sequence of eigenvalues λ is found by inserting the numerical values of κz

(or δ) in (38). The solvability condition for the eigenvalue problem is found by noticing that boundaryconditions (33) and (36) imply that both Zz(z) and Z ′

z(z) must be 0 at z = −1 and z = 1. For Zz(z)not to be 0, the determinant formed by the system of these four conditions S1, must be 0, whichprovides a first solvability condition:

(

δκz tanh(1

δ) − tan(κz)

) (

δκz tan(κz) + tanh(1

δ)

)

= 0. (40)

The system (40,39) admits exactly one solution for (κz , δ) in each interval [qπ/2, (q+1)π/2]×]0,∞[ withq ∈ Z − 0. Modes defined this way are called Orr-Sommerfeld eigenmodes by analogy with stabilityproblems [36]. Since (40) expresses as the product of two terms, there are two types of Orr-Sommerfeldmodes. Modes of the first type satisfy

δκz tanh(1

δ) − tan(κz) = 0 (41)

and are found by solving the system formed by boundary conditions (33,36) for Zz and (33,35) for Zx

and Zy:

eOSsp,0 (λ) =

iκzk⊥

(

cos(κzz)

cos(κz)− cosh( z

δ )

cosh(1δ )

)

+ k2⊥ez

(

sin(κzz)

sin(κz)− sinh( z

δ )

sinh(1δ )

)

exp(ik⊥ · r⊥). (42)

For these modes, the velocity in the x−y plane has a profile along z that is symmetric about the planez = 0. We call them symmetric modes for short. They are found in intervals κz ∈ [qπ/2, (q + 1)π/2]for which q is even. Similarly, Orr-Sommerfeld modes of the second type are anti-symmetric and arefound in such intervals where q is odd. They satisfy

δκz tan(κz) + tanh(1

δ) = 0 (43)


and are expressed as:

eOSap,0 (λ) =

iκzk⊥

(

sin(κzz)

sin(κz)− sinh( z

δ )

sinh(1δ )

)

− k2⊥ez

(

cos(κzz)

cos(κz)− cosh( z

δ )

cosh(1δ )

)

exp(ik⊥ · r⊥). (44)

If, on the other hand, detS1 6= 0, then Zz(z) = 0 for the corresponding mode, which is then calleda Squire mode. For Squire modes not to be trivial, the determinant of the system S2 formed by theremaining four boundary conditions (33,35) must be 0. This defines the Squire family of eigenmodesand eigenvalues:

(

S tanh(1

δ) − M tan(κz)

) (

S tan(κz) + M tanh(1

δ)

)

= 0, (45)

with

S =1

κz

(

1

δ2− k2

⊥ − Ha2

)

and M = δ(

κ2z + k2

⊥ + Ha2)

.

The system (45,39) admits exactly one solution for (κz, δ) in each interval [qπ/2, (q + 1)π/2]×]0,∞[with q ∈ Z. Here again, since the determinant detS2 expresses as a product, Squire modes are eithersymmetric or anti-symmetric. Symmetric Squire modes satisfy:

S tan(κz) + M tanh(1

δ) = 0 (46)

and

eSsp,0(λ) = k⊥ × ez

(

cos(κzz)

cos(κz)− cosh( z

δ )

cosh(1δ )

)

exp(ik⊥ · r⊥). (47)

Here again, symmetric eigenmodes are found for κz ∈ [qπ/2, (q+1)π/2] in intervals for which q is even.Anti-symmetric modes, on the other hand are found in such intervals where q is odd. They satisfy:

S tanh(1

δ) − M tan(κz) = 0 (48)

and

eSap,0(λ) = k⊥ × ez

(

sin(κzz)

sin(κz)− sinh( z

δ )

sinh(1δ )

)

exp(ik⊥ · r⊥). (49)

An important difference between Squire and Orr-Sommerfeld modes, apart from the fact that Squiremodes have no component along ez, is that some Squire modes satisfy 0 < κz < π/2. At high Hart-mann numbers, these modes exhibit only very little dependence along ez outside of the boundary layerof thickness δ. Actual flows with such characteristic are called quasi-two-dimensional and are knownto appear at high Hartmann numbers. Clearly, the Squire modes with 0 < κz < π/2 are perfectlysuited to represent these flows. Examples of profiles along ez from the hierarchy of Orr-Sommerfeldand Squire modes are given in figure 2 and 3.A further similarity with real flows is that for low values of |λi|, δ ≃ Ha−1 to a very good approxima-tion, as in the real laminar Hartmann layer. On the other hand, for high values of |λi|, the values ofδ are much more scattered, with some ”purely viscous” modes and other modes with a much thickerboundary layer thickness. A combination of such modes would certainly represent efficiently the doubledeck structure of the turbulent Hartmann layer theorised by [1] (see [31] for more details).Finally, the system ep,0(λ) found in this section is clearly orthogonal, so it can be used in place ofǫp,0(λ) in projections involving (16). Further normalization would be required to obtain an orthonor-mal system. Since, however, we shall only need orthogonality to illustrate our purpose, we shall skipthis step for the sake of clarity.


Ha = 0.001q = 0 q = 1 q = 6

ORR −1 −0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

z

−1 −0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

z−6 −4 −2 0 2 4 6

−1

−0.5

0

0.5

1

z

SQ 0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

z

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1z

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

zHa = 100

q = 0 q = 1 q = 6

ORR −1 −0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

z

−0.5 0 0.5 1

−1

−0.5

0

0.5

1

z

−6 −4 −2 0 2 4 6

−1

−0.5

0

0.5

1

z

SQ 0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

z

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

z

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

z

Fig. 2 z-profile of the three components of the eigenmodes for Ha = 0.001, 100. Dotted: Zz(z)/k2⊥, solid line:

Zx(z)/(ikx) = Zy(z)/(iky) for Orr-Sommerfeld modes (ORR) and Zx(z)/ky = −Zy(z)/kx for Squire modes(SQ).


Ha = 5000q = 0 q = 1 q = 6

ORR −1 −0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

z

−0.5 0 0.5 1

−1

−0.5

0

0.5

1

z−5 0 5

−1

−0.5

0

0.5

1

zSQ 0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

z

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

z

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

zFig. 3 z-profile of the three components of the eigenmodes for Ha = 5000. Dotted: Zz(z)/k2

⊥, solid line:Zx(z)/(ikx) = Zy(z)/(iky) for Orr-Sommerfeld modes (ORR) and Zx(z)/ky = −Zy(z)/kx for Squire modes(SQ). For such a high Hartmann number, the boundary layers at z = 1 and z = −1 are hardly noticeable. Inparticular, the region near the wall where Z′

z approaches 0 is too small to appear on the graph.

4.3 Associated eigenvectors

In order to complete the canonical system over which the velocity field is expanded, we now need tofind possible associated eigenvectors for the eigenvectors found in the previous subsection. In principle,the possible existence of associated eigenvectors is checked by solving the eigenvalue problem from theadjoint operator L†. Associated eigenvectors can be found if and only if the system of eigenvectorsand adjoint eigenvectors is not biorthogonal. In the frame of the present work however, we wish todemonstrate the efficiency of our spectral approach on an example that is simple enough not to requireusing the adjoint problem (see section 4.4). Since ep,0(λ) has been found to be orthogonal, the exactexpression of the adjoint system ǫp,0(λ) is indeed not required, so we shall therefore skip this stepand instead only look for possible associated eigenvectors by solving the sequence of equations (13)until no more solutions can be found. Starting from any eigenvector ep,0 found in section 4.2, we lookfor the first associated vector by solving for ep,1 the inhomogeneous differential system:

(L − λI)ep,1 = −ep,0, (50)

with boundary conditions (33),(35) and (36). ep,1 is therefore of the form:

ep,0 =(

A1 cos(κzz) + A2 sin(κzz) + A3 exp(z/δ) + A4 exp(−z/δ) + Z(part)(z))

exp(ik⊥ · r⊥) (51)

=(

Z(homo)(z) + Z(part)(z))

exp(ik⊥ · r⊥),

where Z(homo) is solution of the associated homogeneous system and Z(part) is a particular integral of(50). For Symmetric OS modes, (33) and (36), imply that Az is solution of S1(A1z , A2z , A3z, A4z)

T =


−(Z(part)z (1), Z

(part)z (−1), Z

(part)′z (1), Z

(part)′z (−1))T , where

Z(part)z (z) = c1z cos(κzz) + 2c3z cosh(

z

δ), (52)

with coefficients

c1 = − (k2⊥ + κ2

z)

2κz

[

κ2z + 1

δ2

] and c3 =(k2

⊥ − 1δ2 ) sin(κz)

4 1δ (κ2

z + 1δ2 ) sinh(1

δ ). (53)

Since detS1 = 0, this system has a unique solution only if the solvability condition

1

δZ(part)

z (1) − tanh(1

δ)Z(part)′

z (1) = 0 (54)

is satisfied on the top of (39) and (41). A similar solvability condition can be found for antisymmetricOS modes and Squire modes as well as symmetric Squire modes, see Table 1. With the notations

c4 =(k2

⊥ − 1δ2 ) cos(κz)

4 1δ (κ2

z + 1δ2 ) cosh(1

δ ),

Gs(z) = −(k2⊥ + λ)

∫

Z(part)x (z)dz

= −(k2⊥ + λ)

(

− c1

κ2z

sin(κzz) +c1

κzz cos(κzz) + 2c4δz cosh(

z

δ) − 2c4δ

2 sinh(z

δ)

)

,

Ga(z) = − tanh(1

δ)(k2

⊥ + λ)

∫

Z(part)x (z)dz

= − tanh(1

δ)(k2

⊥ + λ)

(

c1

κ2z

cos(κzz) +c1

κzz sin(κzz) + 2c3δz sinh(

z

δ) − 2c3δ

2 cosh(z

δ)

)

.

(55)

Mode Solvability condition Particular integral

Sym. OS 1δZ

(part)z (1) − tanh( 1

δ)Z

(part)′z (1) = 0, Z

(part)z (z) = c1z cos(κzz) + 2c3z cosh( z

δ)

Anti. OS 1δ

tanh( 1δ)Z

(part)z (1) − Z

(part)′z (1) = 0, Z

(part)z (z) = −c1z sin(κzz) + 2c4z sinh( z

δ)

Sym. S M tanh( 1δ)Z

(part)x (1) − Z

(part)′x (1) = Gs(1), Z

(part)x (z) = −c1z sin(κzz) + 2c4z sinh( z

δ)

Anti. S MZ(part)x (1) − tanh( 1

δ)Z

(part)′x (1) = Ga(1), Z

(part)x (z) = c1z cos(κzz) + 2c3z cosh( z

δ)

Table 1 Mode families and corresponding solvability conditions.

We have checked for the existence of associated eigenvectors for Ha = 0.001, 100, 5000 and (kx, ky) ∈1...1002 and found none. Should cases arise where ep,1(λ) eigenvector exists, a possible second asso-ciated eigenvector ep,2(λ) would have to be sought for by solving the next equation in the hierarchy(13) with boundary conditions (33),(35) and (36). The process would have to be repeated up to rankMp for which no solution can be found.

4.4 Square array of quasi-two-dimensional vortices in a channel

As announced in introduction, one of the main advantages of using the sequence of the least dissipativemodes is that the resolution of the Hartmann layer incurs no additional computational cost. We shallnow illustrate this fact on a simple example in a channel flow geometry. For the configuration defined atthe beginning of this section, let us assume that the flow is in a laminar state for which the interaction

parameter Ha2

Reis large enough to ensure that inertia is negligible compared to the Lorentz Force, and

so that the flow is in a laminar state and the quadratic terms in (3) can be neglected. We then apply


a quasi-two-dimensional periodic force on the flow in order to reproduce the square array of vorticesstudied by [40]:

f =

sin x cos y− cosx sin y

0

(

cos(κzz)

cos(κz)− cosh( z

δ )

cosh(1δ )

)

, (56)

where κz and δ are solutions of (39, 46) for q = 0 and k2⊥ = 2. The solution for the velocity field in

steady regime is obtained by direct application of (30), which, in the absence of associated vectors canbe simplified as:

up,0(λ) =fp,0(λ)

λ, fp,0(λ) =

(

ep,0(λ), f)

L2

, (57)

where ep,0(λi) spans the linear subspace that contains f and therefore satisfy fp,0(λi) = (ep,0(λi), f) 6=0. The only vectors ep,0(λi) satisfying this condition are symmetric Squire modes of the form (47)with q = 0 and k⊥ ∈ (1, 1)T , (−1,−1)T , (−1, 1)T , (1,−1)T. Their exact expression is obtained bydetermining the values of λ from (38), so the final solution can be written:

u(x, y, z) =

4∑

p=1

up,0(λ)ep,0(λ) = 2π2I

λ

−110

sin(x + y) −

110

sin(x − y)

(

cos(κzz)

cos(κz)− cosh( z

δ )

cosh(1δ )

)

,

(58)

where

I =

1∫

−1

(

cos(κzz)

cos(κz)− cosh( z

δ )

cosh(1δ )

)2

dz. (59)

For Ha = 1000, λ = −1002, κz = 0.0447 and δ = 0.001.As predicted from the properties of ep,0(λ), the full three-dimensional solution is obtained with asingle profile along z representing the Hartmann layer profile and only modes present in the forcing.In contrast, a full 3D solution based on Tchebychev polynomials at Ha = 1000 is beyond the reach ofcurrent common computers.

4.5 Nonlinear and turbulent flows

We now turn our attention to the fully non-linear equation (3), with the geometry and boundaryconditions defined in this section. A full resolution would take us well beyond our scope of simplyexplaining our spectral method as it would require a full numerical implementation. We shall howeverpoint out the steps involved by such a numerical resolution and underline its main differences withmore traditional spectral methods.The treatment of the non-linear terms carries the main difficulty. In many pseudo-spectral codes, atevery time step, they are reconstructed in real space and then transformed back into spectral space.This involves an inverse transformation of the velocity field and a direct transformation of the nonlinearterms. The existence of fast algorithms to calculate these transformations when the working basis iseither that of Fourier modes or Tchebychev polynomials is one of the reason for their popularity. Thesame approach can be applied when working with the canonical system of L: u is first reconstructed byapplying the inverse multi-functional transform T −1 defined by (18) to coefficients up,m found at theprevious iteration. The coefficients Bp,m(λi, t) at the current iteration are then obtained by calculatingthe multi-functional transform T defined by (16) of u · ∇u. This requires the knowledge of the systemof adjoint eigenvectors and associated vectors (ǫp,m). Clearly, such a treatment of the nonlinear termswould turn out to be more costly than in traditional Tchebychev-based pseudo spectral methods, sosome part of the computational cost saved by using the canonical system of L would be lost there. Analternate, less costly method would consist of treating the nonlinear term in the spectral space only.The second important question is that of knowing which modes should be used in the expansion. Using


the canonical system for the linear problem has allowed us to write the problem under the form of atriangular system were only the modes involved in the expression of the forcing f were involved. Thepicture is different when handling the full nonlinear equation as the quadratic terms generate additionalmodes. The number of modes required to resolve the flow completely can be evaluated heuristicallyor by finding an upper bound for the attractor dimension of the underlying dynamical system. Wehave shown previously that it scales as Re2/Ha. We are then left with the choice of which modesshould be used, or equivalently, we need to sort the elements of the canonical system in a ”coarse” to”fine” order. In this regard, L has real, negative eigenvalues λi only. Since these represent the lineardecay rate of their related eigenmodes and associate vectors, it is natural to sort them by ascendingvalue of −λi. Low values of −λi are associated to large, less dissipative flow structures or that varyless along the field direction, and higher values of −λi, to smaller, more dissipative ones, or that varymore along the field direction. The highest values of −λi required for the full resolution relates to theequivalent of the Kolmogorov scale for the (anisotropic) MHD channel flow. This method is in fact nodifferent to the one that has been applied in pseudo-spectral numerical simulations of hydrodynamicturbulence for decades. In three-dimensional spatially periodic domains, Fourier modes are sortedaccording to the norm of their eigenvector ‖k‖. Since for Ha = 0, −λ = k2,

√−λ may be interpreted

as a generalisation of k that reflects the anisotropy of MHD turbulence. Practically speaking, themodes that need to be taken into account to resolve the flow completely are found by taking all modeswith −λ(kx, ky, κz) < −λmax such that the volume thus defined in the (kx, ky, κz) space encompassesRe2/Ha modes. A schematic representation of the modes involved is shown in figure 4. The propertiesof the families of curves defined by λ = const. are rich and we have previously shown that they recoverthe important features of MHD turbulence known from other experimental and theoretical studies,among which the size of the smallest scales and the fact that for high values of Ha2/Re, the modesare confined outside of the so-called Joule cone, centred on the ez axis in the (kx, ky, κz)-space [2,31].Finally, it should be mentioned that we have been able to apply this very approach to the case of arectangular domain with periodic boundary conditions in the three dimensions of space. It was foundthat far away from where the forcing was located in the spectral space, the lines of constant energytended to coincide with the iso-λ curves, thus confirming the relevance of λ as spectral parameter forMHD turbulence [32].

4.6 Transition to three-dimensionality in quasi-two dimensional flows

Because the derivation of the sequence of least dissipative mode requires many, and even most ofthe physical ingredients from the problem of interest, they offer far more than an effective way ofrepresenting the flow. As far as MHD channel flows are concerned, we have seen that both the anisotropyof the structures and the computationally costly Hartmann boundary layers were to a large extendrepresented in the sequence. We shall now see that the transition between quasi-two-dimensional andthree-dimensional flows is just as well contained in the properties of the sequence.

Let us assume a flow that is forced only by a combination of Squire modes with q = 0. This repre-sents a quasi-two-dimensional forcing such as that in the experiments of [37,38,8]. We shall now studythe conditions of appearance of three-dimensionality in such a flow. The first point to notice is thatstrictly speaking, none of the least dissipative modes is truly quasi-two-dimensional in the sense that inthe limit Ha → ∞, the velocity field would be strictly invariant along z outside the Hartmann layers.We have indeed shown previously that the first modes in the sequence of least dissipative modes wereSquire modes with 0 < κz < π/2, and k⊥ > 0. It can be seen from (47) that these exhibit a small,symmetric modulation along the magnetic field. This means that three-dimensionality appears firstunder this particular form rather than under that of vortex disruption, as in problems with periodicboundary conditions (see [30] for the corresponding studies on the sequence of least dissipative modesand [41] for DNS). We have previously been able to explain this effect by means of asymptotic theory[33] and it was observed numerically by [29] who called it ”barrel” or ”cigar” effect. We showed thatvorticity in the core flow which is indeed present as soon as k⊥ ≥ 0 induced eddy currents closing in theHartmann layer that, in turn were responsible for this weak three-dimensionality, where the flow wasstratified. Our asymptotics, however assumed that the Hartmann layer thickness remained constantwhile figure 3 in [31] clearly shows that it varies slightly as soon as inertia appears. The sequence ofleast dissipative modes then holds the remarkable property that only flows free of vorticity along ez


Determining Modes

Non−determiningModes

k⊥

κz λ = λmax

0

(a) (b) (c)

ex

eyez

Fig. 4 A representation of the modes required to resolve a turbulent MHD channel flow. Required modes arerepresented in gray. The line λ = λmax that marks the border of this domain indicates the most dissipativescales present in the flow. Three dimensional plots show the contours of u.ez for the Orr-Sommerfeld modesthat correspond to the values of (kx, ky , κz) located just below this curve, that can therefore be thought of asthe anisotropic MHD equivalent to the Kolmogorov scales.

can be strictly quasi-two-dimensional. We have recently studied experimentally a square array of elec-trically driven vortices in a channel under transverse magnetic field [20] similar to those from section4.4. There too, we found that even at high magnetic field, three-dimensionality was always presentunder the form of differential rotation in individual vortices. This confirms that the modulation alongz carried by the least dissipative modes corresponds to a fine property of the real flow.

Stronger three-dimensionality appears with modes such that q ≥ 1 as soon as the λ = λmax graphencloses a value of κz ∈ [π/2, π], for which the flow cannot be described as stratified anymore. Squiremodes with κz in this interval are antisymmetric modes so the corresponding profile of horizontalvelocity exhibits a strong deformation along z, but no vertical velocity. We have previously definedthis type of three-dimensionality as morphological [32]. Orr-Sommerfeld modes with κz ∈ [π/2, π] arealso antisymmetric and carry both morphological and dynamical three-dimensionality, which is char-acterised by a non-zero velocity along the magnetic field lines. The first Squire (resp. Orr-Sommerfeld)mode in the sequence of least dissipative modes therefore correspond to the appearance of two dif-ferent types of three-dimensionality in the flow. We shall call their respective wavevectors (kα

⊥, καz )

and (kβ⊥, κβ

z ). These are found by noticing from (38), that the maximum value of κz is reached for

κ2z/(k2

⊥ + κ2z) = −λ/(2Ha2). Then, since π/2κz(q = 1) < π, both kα

⊥ and kβ⊥ must satisfy:

√

Haπ

2+

π2

4< k⊥ <

√

Haπ + π2, (60)


10−2

100

102

101

102

Ha

k3

D⊥

Orr−Sommerfeld, k⊥β

Squire, k⊥α

10−2

100

102

102

103

104

Ha

|λ3

D|

Orr−Sommerfeld

Squire

Fig. 5 Variations with Ha of the lengthscales of the first three-dimensional Orr-Sommerfeld and Squire modes(left) and corresponding values of |λ3D| (right).

which provides the scaling kα⊥ ∼ Ha1/2 and kβ

⊥ ∼ Ha1/2. The exact value of kα⊥ (resp. kβ

⊥) is found bysolving (39, 43) (resp. (39, 48)) with q = 1 and k⊥ spanning all integer values in the interval (60). Among

the solution obtained, kα⊥ and kβ

⊥ correspond to the modes with the lowest values of λ3D given by (38).The respective lengthscales of the corresponding modes and corresponding values of λ3D are shown onfigure (5). Three remarkable features appear: firstly, for Ha < 50 both morphological and dynamical

three-dimensionalities appear at the same scale kα⊥ = kβ

⊥ = k3D⊥ . Secondly, for all values of Ha, the

first three-dimensional Orr-Sommerfeld mode (kβ⊥, κβ

z ) produces less dissipation than the first three-dimensional Squire (kα

⊥, καz ) mode. This implies that dynamical three-dimensionality, that is carried by

Orr-Sommerfeld modes appears simultaneously with morphological three-dimensionality, that is onlycarried by both Orr-Sommerfeld and Squire modes. In other words, motion along the magnetic fieldlines appears as soon as strong three-dimensionality is present in the flow. The velocity contours of thefirst three-dimensional Orr-Sommerfeld and Squire modes for Ha = 3163 are represented on figure 7,illustrating that they have different transverse wavelength k⊥. Thirdly, as soon as Ha exceeds 50 thefirst 3D Orr-Sommerfeld mode appears at a larger scale then the first 3D Squire mode. Furthermore, atlow Ha, the first 3D Squire mode and the first Orr-Sommerfeld mode follow each other in the sequenceof least dissipative modes. When Ha increases, however, more and more Orr-Sommerfeld modes arepresent before the first 3D Squire mode appears (the exact number is reported on figure 6). Since for

Ha > 50, kα⊥ > kβ

⊥, this indicates that at high Ha, vortex disruption appears at kα⊥ while a second

transition happens for higher flow intensities that sees smaller scales k > kβ⊥ exhibit a more complex

three-dimensional behaviour, which would have to be studied by a full DNS using the least dissipativemodes.

5 Conclusion

We have presented a spectral method for the resolution of linear and non-linear PDEs that can useany couple of biorthogonal bases. For low Rm-MHD channel flows, we have obtained these bases asthe prime and adjoint canonical systems of the operator that represents the sum of viscous and Jouledissipation in the governing equations, together with the problem’s boundary conditions. Because thesemodes are derived from an operator that reflects the physics of this class of flow, the set of modes builtthis way is made of elements that correspond to structures of the actual flows. In particular, laminarand turbulent Hartmann boundary layers that develop along the channel walls appear as built-in fea-tures and therefore do not incur any specific computational cost, as they do in spectral methods based


10−4

10−2

100

102

104

101

102

103

Ha

nu

mb

er

of

mo

de

s

Fig. 6 Number of Orr-Sommerfeld modes which are present when the first 3D Squire mode appears.

ex

eyez

Fig. 7 Slices of the velocity contours of the first three-dimensional Orr-Sommerfeld mode (left) and the first

three-dimensional Squire mode (right) for Ha = 3163. One clearly sees that kα⊥ > kβ

⊥.

on Tchebychev polynomials or as in alternate numerical methods such as finite difference methods orfinite volume methods. In fact, we have shown that for linear flows, a simple solution could be found,that required only the modes imposed by the application of an external forcing. If this forcing consistsonly of a few modes, as when driving a quasi-two dimensional periodic array of vortice [38],[40] thenthese few modes are enough to express the velocity field, and this for arbitrary high Hartmann numbers.

Apart from this method, only wall functions currently make it possible to calculate high Hartmannflows without suffering from the limitations imposed by meshing the Hartmann layer. They consist ofboundary conditions obtained either from asymptotic theory [34] or empirical modelling that apply atthe interface between the boundary layer and the outer flow. Although their implementation involvessome numerical cost [24,29], they can be adapted to a wide class of problems and combined to virtuallyany numerical technique. By contrast, implementing the sequence of least dissipative modes in spectralmethods requires first to find them, which in complex geometries may only be possible numerically.Once these are known however, the same set can be used for time-dependent simulations at any flow


intensity. The main benefit of this technique over that of wall functions is that it implies no assumptionon the flow, and therefore seems the only method currently giving any hope of performing DNS athigh Hartmann numbers (note that spectral methods based on the least dissipative modes could alsobe combined with modelling techniques such as Large Eddy Simulations as in [22] to further reducecomputational costs). Having a fully resolved Hartmann layer is of crucial importance in flows whereit is strongly disturbed by inertia, as in wakes [29,9,10] and bends [27,39]. Asymptotic wall functionssuch as those from [34] indeed fail to represent such a complex dynamics.The least dissipative modes offer also far more than a convenient way of modelling the flow. We havefound that since their properties reflect those of the flow closely, they could serve as a fine predictivetool: in particular they allowed us to confirm previous experimental and theoretical findings accord-ing to which three-dimensionality first appeared under a ”weak” form leading to the deformation ofcolumnar vortices into barrels [29,33,21]. We could also show that when ”strong” three-dimensionalityappears, i.e. when the flow looses its stratified structure, then motion along the magnetic field is ig-nited. This means that when vorticity is present in the flow, it is either in a weakly three-dimensionalstate (stratified), or strongly three-dimensional, with three-dimensional motion, in contrast with flowsin periodic domains that can be strictly 2D, or exhibit morphological three-dimensionality withoutmotion along the field lines [32].Clearly, one of the main motivations for developing such a method is to calculate turbulent flows, andthis is now the main focus of our attention. In this regard, lifting the need of meshing the Hartmannlayers implies that the number of modes required for a full resolution of the flow using our new basis isnot more than that strictly required by the flow’s dynamics, of the order of Re2/Ha according to ourprevious estimates [30,31,32]. This means that with a relatively low computational cost of 106 modes,flow parameters such as Ha = 100, and Re = 104, or Ha = 1000, and Re = 3.104 or even Ha = 104,and Re = 105 could be fully resolved, without any modelling.Beyond turbulence and MHD, this approach can clearly benefit many other types of flows and even ofPDE-governed problems. In complex geometries or when anisotropy is present, chances are that thenumber of modes required to resolve the flow fully will be significantly lower than that of Re9/4, thatis only supposed to apply to homogeneous hydrodynamic turbulence. In this case, it may be worthchecking if the the specific structures present in the flow (anisotropic vortice, boundary layers, etc...)can be represented by the elements of the canonical system of the operator obtained from the linearpart of the Navier-Stokes equation with the problem’s boundary conditions. Even if those have to beobtained numerically, the solution of this single time-independent eigenvalue problem can be used tosimulate the time evolution of the system for arbitrary initial conditions.

Acknowledgements We are also grateful to Bernard Knaepen and Oleg Zikanov for the useful discussionsand comments during the MHD 2007 summer program at the Universite Libre de Bruxelles.

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