PACS: 05.60.Cd; 05.45.Df; 47.53.+n; 47.56.+r

1. Introduction

The transport equation for a concentration field, *C*, analyzed in this work is obtained from the conservation law

and a Fourier-Fick’s advective-diffusive closure relation of the form

where **v**(*t*) is a velocity vector field, both functions of time (not space). The resulting transport equation is

Our main interest regarding the structure of the last equation concerns the time-dependence on advection and diffusion, and the anisotropy of the diffusion tensor. This equation and diverse variants of it can be found in many applications, including particle transport in anisotropic time-dependent stochastic velocity fields, where the diffusion tensor is related to velocity correlation functions. Also, tracer or pollutant transport in underground porous formations can be captured within this last approach by associating the stochastic velocity variations to permeability changes of the porous media^{1}^{,}^{2}^{,}^{3}^{,}^{4}^{,}^{5}^{,}^{6} and ^{7}, (ii) transport in periodic porous media^{8}, (iii) transport of contaminants carried by the wind in the atmosphere^{9}^{,}^{10}^{,}^{11}, (iv) low- and high-temperature magnetically confined plasmas, where charged particles can displace rapidly along the magnetic lines but diffuse slowly in the perpendicular (with respect to the magnetic field) direction^{12}^{,}^{13}^{,}^{14}, (v) gas transport in cylindrical industrial ducts^{15}^{,}^{16}, (vi) transport of molecular spins in bio-tissues for nuclear magnetic resonance applications^{17}^{,}^{18}^{,}^{19} and (vii) Brownian particles with memory, where the diffusion tensor’s time-dependence can be related to long correlations of the randomly fluctuating forces acting on the particle^{20}^{,}^{21}.

In this work we obtain an analytical solution of Eq. (3) and explore the implications of the anisotropy on the transport process by examining the natural coordinate systems involved. It is to be mentioned that some practical approaches propose the description of the so called *anomalous transport* via a time-dependent diffusion coefficient (as in Eq. (3)), since the resulting mean square standard deviation can increase above or below the linear time dependence (as we will show later), it is interpreted as *sub-* or *super-diffusion*. As it is well-known, non-diffusive transport also naturally arises in the context of Continuous Time random Walk models with non-Gaussian and/or non-Markovian statistics see for example^{22}^{,}^{23} and references therein. However, the present work focuses on a different source of non-diffusive behavior. In particular, we are interested in systems that despite being described by Fourier-Fick’s closure relations exhibit non-diffusive transport because of the anomalous time dependence and anisotropy in the diffusion tensor.

To formulate the anisotropic transport problem on a general setting let *i* = 1*, … , N*, denote a normalized basis of vectors in

Notice that the *i*-th component *q*
_{
i
} is not necessarily equal to the projection of vector **q** onto the basis vector **f**
_{
i
} , *i.e.*, , *q*
_{
i
} are contravariant components while

In this work we restrict attention to homogeneous anisotropic media, such that the diffusion tensor, ^{2}^{,}^{4}^{,}^{24}, however, we keep the time dependence in both, the diffusion tensor, **v** = **v**(*t*). Furthermore, we focus on a particular class of systems characterized by the condition that their diffusion tensor can be brought to a diagonal form (with time-dependent elements), when written in the coordinate system corresponding to the basis *SO*(*N*) group, and thus

2. Model

To construct the model we assume that *u*-coordinate system, *i.e.*,

Substituting Eq. (5) into Eq. (1), and using

or equivalently,

Note that in the *i.e.*, it has no cross terms in the directional derivatives

Clearly, if a different coordinate system is adopted, cross terms (involving the directional derivatives along the corresponding coordinate directions) will in general appear. As a particular important example, consider a Cartesian coordinate system **e**
_{i}, *i* = 1*, … , N*. In this case,

where the constants

define the change of bases with the corresponding change of coordinates given by the linear transformation

With inverse

Notice that the metric tensor can be written as

In the coordinates

where

Note that, by construction,

When written in the

If the ansatz *N* equations:

with *i* = 1*, … , N*, in the sense that any solution of (15) is also a solution of (7). Along this work there will be sum over repeated indices only when explicitly stated by the summation symbol. Furthermore, by introducing the temporal variables

(15) reduces to

for each *i* = 1*, … , N*. In order for

and the solution of Eq. (12) is

where

To calculate the mean standard deviation of a pulse with respect to the normalized distribution *C*(*u*
_{
i
} , *t*) we define the expectation value *F* as

where the translated coordinate system in Eq. (14) has been used. By noticing that

which means, *i.e.*, the mean pulse position in the **f**
_{
i
} direction drifts at the speed *v*
_{
i
} . The mean square deviation in the **f**
_{
i
} direction,

Provided the diffusion tensor components,

3. Interpretation

When the **f**
_{
i
} basis is orthonormal (*i.e.*, Cartesian), *q*
_{
i
} and **f**
_{
i
} direction depends only on the gradient of *C* along that same direction. However, when *q*
_{
i
} depends only on the gradient of *C* along direction **f**
_{
i
} , the projection *C* along the *j*-th direction, for

By choosing *q*
_{
i
} to obey the relation (5) and therefore the

where **d**
_{
i
} and *i*-th eigenvector and the corresponding eigenvalue of the second order tensor

and we remind the reader that

From expression (24) it is clear that if

Summarizing, in general there are three different basis (see Fig. 1): (i) the non orthonormal constant basis in terms of which the problem takes its simplest form, *N =* 2 for the sake of simplicity.

4. Applications

Let us first consider a general two-dimensional anisotropic situation. We consider a non-orthogonal basis spanned by the vectors *i* = 1, 2, as shown in Fig. 1, with

In such a basis, we propose a diffusion tensor whose components are written as

According to (13) the diffusion tensor written in Cartesian coordinates takes the form:

and its eigenvalues and eigenvectors are:

respectively, where *n*
_{
i
} is a normalization factor. Consistent with the general discussion in Sec. 3, the eigenvectors **d**
_{
i
} are orthonormal and different from **
f
**

_{i}. The vector

**d**

_{1}corresponds to the direction of maximum diffusion (since

**d**

_{1}. However, for the situation described below in cases 4.1 and 4.2, it can be shown that for large values of time

*T*)

4.1. Anisotropic Sub-Diffusion

As a first example let us consider a case in which sub-diffusion takes place. To this end we provide an adequate form of

where *T* is the system’s characteristic decay correlation time. The behavior in time of

The corresponding mean square displacement is,

for

For short times,

which corresponds to sub-diffusion since

From expressions (27), (28) and (29) we know that **d**
_{1} corresponds to the direction of maximum diffusion, and also that, as shown in Fig. 2, it satisfies the asymptotic

4.2. Anisotropic Super-Diffusion

Let us next assume that ^{2}^{,}^{4}, where the diffusion coefficient is the integral of the velocity correlation function over time. If a power law correlation function is considered^{2}^{,}^{3}:

In this case, the mean square displacement takes the form:

For short times,

For large values of *t*
_{
D
} and *i.e.*,

This example has been previously studied by Numbere and Erkal^{3} by solving the corresponding transport equation, Eq. (12), numerically. Our analytical results confirm to certain extent their findings. In particular, for

The explicit form of the concentration pulse *C*(*x, y, t*) can be found directly through expression (19). In Fig. (6) we show the time evolution of the concentration profile for three sets of parameter values **d**
_{1} which depends on the ratio **f**
_{2}, while for **f**
_{1} + **f**
_{2}.

4.3. Hydrodynamic Dispersion

Let us finally analyse the case of hydrodynamic dispersion in an homogeneous isotropic porous media, where anisotropic dispersion appears intrinsically. The anisotropy exhibits a maximum in the velocity direction and the minimum in the perpendicular directions. In this case, the dispersion tensor is written as^{25}^{,}^{26}

where **f**
_{1} in the direction of the velocity (direction of maximum dispersion). Thus

and Eq. (5) gives

Thus the model of the previous subsections can be applied with *a*
_{
ij
} is a rigid rotation of angle

and

Non-diffusive transport can be introduced in this model by assuming that *t* conveniently chosen as it was done in the previous examples. The solution for the tracer concentration **d**
*i* = **f**
*i*.

5. Summary and Conclusions

In this paper we have analytically solved an *N*-dimensional advection-diffusion equation with time-dependent coefficients and diffusion anisotropy. The problem has been formulated in terms of a non-orthogonal basis in order to accommodate anisotropy. There appear naturally two basis vector: the non-orthonormal constant basis in terms of which the problem takes its simplest form, **f**
_{1}, **f**
_{2} or **f**
_{1} + **f**
_{2}, depending on the value of the ratio

We have particularly discussed the advection-diffusion equation in the context of its application to particle or solute transport in stochastic velocity fields. Here the diffusion tensor is given in terms of a time integral of the correlation function. The resulting transport behavior can display features of the so called sub- or super-diffusion, where the standard deviation increase in time with a power lower or lager the the unity, respectively. A known case that consider power law anisotropic dispersion coefficients^{2}^{,}^{3}^{,}^{24}^{,}^{25} was examined, and we analytically reproduced the numerical results they obtained^{3}.

Finally, an interesting direction of future work might be to extend this model to the domain of fractional diffusion, where non-locality (in space and time) naturally arises.