Mathematical descriptions of the electromagnetic field

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There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

Vector field approach

Main article: Classical electromagnetism

The most common description of the electromagnetic field uses two three-dimensional vector fields called the electric field and the magnetic field. These vector fields each have a value defined at every point of space and time and are thus often regarded as functions of the space and time coordinates. As such, they are often written as E(x, y, z, t) (electric field) and B(x, y, z, t) (magnetic field).

If only the electric field (E) is non-zero, and is constant in time, the field is said to be an electrostatic field. Similarly, if only the magnetic field (B) is non-zero and is constant in time, the field is said to be a magnetostatic field. However, if either the electric or magnetic field has a time-dependence, then both fields must be considered together as a coupled electromagnetic field using Maxwell's equations.

Maxwell's equations in the vector field approach

Main article: Maxwell's equations

The behaviour of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or electrodynamics (electromagnetic fields), is governed by Maxwell-Heaviside's equations:

Maxwell's equations (vector fields)
${\displaystyle \mathrm{\nabla }\cdot \mathbf{E}=\frac{\rho }{{\epsilon }_0}}$	Gauss's law
${\displaystyle \mathrm{\nabla }\cdot \mathbf{B}=0}$	Gauss's law for magnetism
${\displaystyle \mathrm{\nabla }\times \mathbf{E}=-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}}$	Faraday's law of induction
${\displaystyle \mathrm{\nabla }\times \mathbf{B}={\mu }_0\mathbf{J}+{\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}}$	Ampère-Maxwell law

where ρ is the charge density, which can (and often does) depend on time and position, ε₀ is the electric constant, μ₀ is the magnetic constant, and J is the current per unit area, also a function of time and position. The equations take this form with the International System of Quantities.

When dealing with only nondispersive isotropic linear materials, Maxwell's equations are often modified to ignore bound charges by replacing the permeability and permittivity of free space with the permeability and permittivity of the linear material in question. For some materials that have more complex responses to electromagnetic fields, these properties can be represented by tensors, with time-dependence related to the material's ability to respond to rapid field changes (dispersion (optics), Green–Kubo relations), and possibly also field dependencies representing nonlinear and/or nonlocal material responses to large amplitude fields (nonlinear optics).

Potential field approach

Many times in the use and calculation of electric and magnetic fields, the approach used first computes an associated potential: the electric potential, ${\displaystyle \phi }$, for the electric field, and the magnetic vector potential, A, for the magnetic field. The electric potential is a scalar field, while the magnetic potential is a vector field. This is why sometimes the electric potential is called the scalar potential and the magnetic potential is called the vector potential. These potentials can be used to find their associated fields as follows: $${\displaystyle \mathbf{E}=-\mathrm{\nabla }\phi -\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}}$$ $${\displaystyle \mathbf{B}=\mathrm{\nabla }\times \mathbf{A}}$$

Maxwell's equations in potential formulation

These relations can be substituted into Maxwell's equations to express the latter in terms of the potentials. Faraday's law and Gauss's law for magnetism (the homogeneous equations) turn out to be identically true for any potentials. This is because of the way the fields are expressed as gradients and curls of the scalar and vector potentials. The homogeneous equations in terms of these potentials involve the divergence of the curl ${\displaystyle \mathrm{\nabla }\cdot \mathrm{\nabla }\times \mathbf{A}}$ and the curl of the gradient ${\displaystyle \mathrm{\nabla }\times \mathrm{\nabla }\phi }$, which are always zero. The other two of Maxwell's equations (the inhomogeneous equations) are the ones that describe the dynamics in the potential formulation.

Maxwell's equations (potential formulation)

${\displaystyle {\mathrm{\nabla }}^2\phi +\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\mathrm{\nabla }\cdot \mathbf{A}\right)=-\frac{\rho }{{\epsilon }_0}}$

${\displaystyle \left({\mathrm{\nabla }}^2\mathbf{A}-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathbf{A}}{\mathrm{\partial }{t}^2}\right)-\mathrm{\nabla }\left(\mathrm{\nabla }\cdot \mathbf{A}+\frac{1}{c^2}\frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}\right)=-{\mu }_0\mathbf{J}}$

These equations taken together are as powerful and complete as Maxwell's equations. Moreover, the problem has been reduced somewhat, as the electric and magnetic fields together had six components to solve for. In the potential formulation, there are only four components: the electric potential and the three components of the vector potential. However, the equations are messier than Maxwell's equations using the electric and magnetic fields.

Gauge freedom

These equations can be simplified by taking advantage of the fact that the electric and magnetic fields are physically meaningful quantities that can be measured; the potentials are not. There is a freedom to constrain the form of the potentials provided that this does not affect the resultant electric and magnetic fields, called gauge freedom. Specifically for these equations, for any choice of a twice-differentiable scalar function of position and time λ, if (φ, A) is a solution for a given system, then so is another potential (φ′, A′) given by: $${\displaystyle {\phi }^{\prime }=\phi -\frac{\mathrm{\partial }\lambda }{\mathrm{\partial }t}}$$ $${\displaystyle {\mathbf{A}}^{\prime }=\mathbf{A}+\mathrm{\nabla }\lambda }$$

This freedom can be used to simplify the potential formulation. Either of two such scalar functions is typically chosen: the Coulomb gauge and the Lorenz gauge.

Coulomb gauge

Main article: Gauge fixing § Coulomb gauge

The Coulomb gauge is chosen in such a way that ${\displaystyle \mathrm{\nabla }\cdot {\mathbf{A}}^{\prime }=0}$, which corresponds to the case of magnetostatics. In terms of λ, this means that it must satisfy the equation $${\displaystyle {\mathrm{\nabla }}^2\lambda =-\mathrm{\nabla }\cdot \mathbf{A}.}$$

This choice of function results in the following formulation of Maxwell's equations: $${\displaystyle {\mathrm{\nabla }}^2{\phi }^{\prime }=-\frac{\rho }{{\epsilon }_0}}$$ $${\displaystyle {\mathrm{\nabla }}^2{\mathbf{A}}^{\prime }-{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2{\mathbf{A}}^{\prime }}{\mathrm{\partial }{t}^2}=-{\mu }_0\mathbf{J}+{\mu }_0{\epsilon }_0\mathrm{\nabla }\left(\frac{\mathrm{\partial }{\phi }^{\prime }}{\mathrm{\partial }t}\right)}$$

Several features about Maxwell's equations in the Coulomb gauge are as follows. Firstly, solving for the electric potential is very easy, as the equation is a version of Poisson's equation. Secondly, solving for the magnetic vector potential is particularly difficult. This is the big disadvantage of this gauge. The third thing to note, and something that is not immediately obvious, is that the electric potential changes instantly everywhere in response to a change in conditions in one locality.

For instance, if a charge is moved in New York at 1 pm local time, then a hypothetical observer in Australia who could measure the electric potential directly would measure a change in the potential at 1 pm New York time. This seemingly violates causality in special relativity, i.e. the impossibility of information, signals, or anything travelling faster than the speed of light. The resolution to this apparent problem lies in the fact that, as previously stated, no observers can measure the potentials; they measure the electric and magnetic fields. So, the combination of ∇φ and ∂A/∂t used in determining the electric field restores the speed limit imposed by special relativity for the electric field, making all observable quantities consistent with relativity.

Lorenz gauge condition

Main article: Lorenz gauge condition

A gauge that is often used is the Lorenz gauge condition. In this, the scalar function λ is chosen such that $${\displaystyle \mathrm{\nabla }\cdot {\mathbf{A}}^{\prime }=-{\mu }_0{\epsilon }_0\frac{\mathrm{\partial }{\phi }^{\prime }}{\mathrm{\partial }t},}$$ meaning that λ must satisfy the equation $${\displaystyle {\mathrm{\nabla }}^2\lambda -{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2\lambda }{\mathrm{\partial }{t}^2}=-\mathrm{\nabla }\cdot \mathbf{A}-{\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}.}$$

The Lorenz gauge results in the following form of Maxwell's equations: $${\displaystyle {\mathrm{\nabla }}^2{\phi }^{\prime }-{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2{\phi }^{\prime }}{\mathrm{\partial }{t}^2}=-{◻}^2{\phi }^{\prime }=-\frac{\rho }{{\epsilon }_0}}$$ $${\displaystyle {\mathrm{\nabla }}^2{\mathbf{A}}^{\prime }-{\mu }_0{\epsilon }_0\frac{{\mathrm{\partial }}^2{\mathbf{A}}^{\prime }}{\mathrm{\partial }{t}^2}=-{◻}^2{\mathbf{A}}^{\prime }=-{\mu }_0\mathbf{J}}$$

The operator ${\displaystyle {◻}^2}$ is called the d'Alembertian (some authors denote this by only the square ${\displaystyle ◻}$). These equations are inhomogeneous versions of the wave equation, with the terms on the right side of the equation serving as the source functions for the wave. As with any wave equation, these equations lead to two types of solution: advanced potentials (which are related to the configuration of the sources at future points in time), and retarded potentials (which are related to the past configurations of the sources); the former are usually disregarded where the field is to analyzed from a causality perspective.

As pointed out above, the Lorenz gauge is no more valid than any other gauge since the potentials cannot be directly measured, however the Lorenz gauge has the advantage of the equations being Lorentz invariant.

Extension to quantum electrodynamics

Canonical quantization of the electromagnetic fields proceeds by elevating the scalar and vector potentials; φ(x), A(x), from fields to field operators. Substituting 1/c² = ε₀μ₀ into the previous Lorenz gauge equations gives:

$${\displaystyle {\mathrm{\nabla }}^2\mathbf{A}-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathbf{A}}{\mathrm{\partial }{t}^2}=-{\mu }_0\mathbf{J}}$$ $${\displaystyle {\mathrm{\nabla }}^2\phi -\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{t}^2}=-\frac{\rho }{{\epsilon }_0}}$$

Here, J and ρ are the current and charge density of the matter field. If the matter field is taken so as to describe the interaction of electromagnetic fields with the Dirac electron given by the four-component Dirac spinor field ψ, the current and charge densities have form: $${\displaystyle \mathbf{J}=-e{\psi }^{†}\mathit{\alpha }\psi \rho =-e{\psi }^{†}\psi ,}$$ where α are the first three Dirac matrices. Using this, we can re-write Maxwell's equations as:

Maxwell's equations (QED)

${\displaystyle {\mathrm{\nabla }}^2\mathbf{A}-\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\mathbf{A}}{\mathrm{\partial }{t}^2}={\mu }_{0}e{\psi }^{†}\mathit{\alpha }\psi }$

${\displaystyle {\mathrm{\nabla }}^2\phi -\frac{1}{c^2}\frac{{\mathrm{\partial }}^2\phi }{\mathrm{\partial }{t}^2}=\frac{1}{{\epsilon }_0}e{\psi }^{†}\psi }$

which is the form used in quantum electrodynamics.

Geometric algebra formulations

Analogous to the tensor formulation, two objects, one for the electromagnetic field and one for the current density, are introduced. In geometric algebra (GA) these are multivectors, which sometimes follow Ricci calculus.

Algebra of physical space

In the Algebra of physical space (APS), also known as the Clifford algebra ${\displaystyle C{\ell }_{3,0}(\mathbb{R})}$, the field and current are represented by multivectors.

The field multivector, known as the Riemann–Silberstein vector, is $${\displaystyle \mathbf{F}=\mathbf{E}+Ic\mathbf{B}=E^k{\sigma }_k+Ic{B}^k{\sigma }_k,}$$ and the four-current multivector is $${\displaystyle c\rho -\mathbf{J}=c\rho -J^k{\sigma }_k}$$ using an orthonormal basis ${\displaystyle \{{\sigma }_k\}}$. Similarly, the unit pseudoscalar is ${\displaystyle I={\sigma }_1{\sigma }_2{\sigma }_3}$, due to the fact that the basis used is orthonormal. These basis vectors share the algebra of the Pauli matrices, but are usually not equated with them, as they are different objects with different interpretations.

After defining the derivative $${\displaystyle \mathbf{\nabla }={\sigma }^k{\mathrm{\partial }}_k,}$$

Maxwell's equations are reduced to the single equation

Maxwell's equations (APS formulation)

${\displaystyle \left(\frac{1}{c}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}}+\mathbf{\nabla }\right)\mathbf{F}={\mu }_{0}c(c\rho -\mathbf{J}).}$

In three dimensions, the derivative has a special structure allowing the introduction of a cross product: $${\displaystyle \mathbf{\nabla }\mathbf{F}=\mathbf{\nabla }\cdot \mathbf{F}+\mathbf{\nabla }\wedge \mathbf{F}=\mathbf{\nabla }\cdot \mathbf{F}+I\mathbf{\nabla }\times \mathbf{F}}$$ from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation. After expanding and rearranging, this can be written as $${\displaystyle \left(\mathbf{\nabla }\cdot \mathbf{E}-\frac{\rho }{{\epsilon }_0}\right)-c\left(\mathbf{\nabla }\times \mathbf{B}-{\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t}-{\mu }_0\mathbf{J}\right)+I\left(\mathbf{\nabla }\times \mathbf{E}+\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\right)+Ic\left(\mathbf{\nabla }\cdot \mathbf{B}\right)=0}$$

Spacetime algebra

Further information: Spacetime algebra § Classical electromagnetism

We can identify APS as a subalgebra of the spacetime algebra (STA) ${\displaystyle C{\ell }_{1,3}(\mathbb{R})}$, defining ${\displaystyle {\sigma }_k={\gamma }_k{\gamma }_0}$ and ${\displaystyle I={\gamma }_0{\gamma }_1{\gamma }_2{\gamma }_3}$. The ${\displaystyle {\gamma }_{\mu }}$s have the same algebraic properties of the gamma matrices but their matrix representation is not needed. The derivative is now $${\displaystyle \mathrm{\nabla }={\gamma }^{\mu }{\mathrm{\partial }}_{\mu }.}$$

The Riemann–Silberstein becomes a bivector $${\displaystyle F=\mathbf{E}+Ic\mathbf{B}=E^1{\gamma }_1{\gamma }_0+E^2{\gamma }_2{\gamma }_0+E^3{\gamma }_3{\gamma }_0-c(B^1{\gamma }_2{\gamma }_3+B^2{\gamma }_3{\gamma }_1+B^3{\gamma }_1{\gamma }_2),}$$ and the charge and current density become a vector $${\displaystyle J=J^{\mu }{\gamma }_{\mu }=c\rho {\gamma }_0+J^k{\gamma }_k={\gamma }_0(c\rho -J^k{\sigma }_k).}$$

Owing to the identity $${\displaystyle {\gamma }_0\mathrm{\nabla }={\gamma }_0{\gamma }^0{\mathrm{\partial }}_0+{\gamma }_0{\gamma }^k{\mathrm{\partial }}_k={\mathrm{\partial }}_0+{\sigma }^k{\mathrm{\partial }}_k=\frac{1}{c}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}}+\mathbf{\nabla },}$$

Maxwell's equations reduce to the single equation

Maxwell's equations (STA formulation)

${\displaystyle \mathrm{\nabla }F={\mu }_{0}cJ.}$

Differential forms approach

See also: Differential form and Exterior algebra

In what follows, cgs-Gaussian units, not SI units are used. (To convert to SI, see here.) By Einstein notation, we implicitly take the sum over all values of the indices that can vary within the dimension.

Field 2-form

In free space, where ε = ε₀ and μ = μ₀ are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential forms is used. The electric and magnetic fields are now jointly described by a 2-form F in a 4-dimensional spacetime manifold. The Faraday tensor ${\displaystyle F_{\mu \nu }}$ (electromagnetic tensor) can be written as a 2-form in Minkowski space with metric signature (− + + +) as $${\displaystyle \begin{array}{rl}\mathbf{F}& \equiv \frac{1}{2}{F}_{\mu \nu }\mathrm{d}{x}^{\mu }\wedge \mathrm{d}{x}^{\nu }\\ & =B_x\mathrm{d}y\wedge \mathrm{d}z+B_y\mathrm{d}z\wedge \mathrm{d}x+B_z\mathrm{d}x\wedge \mathrm{d}y+E_x\mathrm{d}x\wedge \mathrm{d}t+E_y\mathrm{d}y\wedge \mathrm{d}t+E_z\mathrm{d}z\wedge \mathrm{d}t\end{array}}$$ which is the exterior derivative of the electromagnetic four-potential ${\displaystyle \mathbf{A}:}$ $${\displaystyle \mathbf{A}=-\varphi \mathrm{d}t+A_x\mathrm{d}x+A_y\mathrm{d}y+A_z\mathrm{d}z.}$$

The source free equations can be written by the action of the exterior derivative on this 2-form. But for the equations with source terms (Gauss's law and the Ampère-Maxwell equation), the Hodge dual of this 2-form is needed. The Hodge star operator takes a p-form to a (n − p)-form, where n is the number of dimensions. Here, it takes the 2-form (F) and gives another 2-form (in four dimensions, n − p = 4 − 2 = 2). For the basis cotangent vectors, the Hodge dual is given as (see Hodge star operator § Four dimensions) $${\displaystyle \star (\mathrm{d}x\wedge \mathrm{d}y)=-\mathrm{d}z\wedge \mathrm{d}t,\star (\mathrm{d}x\wedge \mathrm{d}t)=\mathrm{d}y\wedge \mathrm{d}z,}$$ and so on. Using these relations, the dual of the Faraday 2-form is the Maxwell tensor, $${\displaystyle \star \mathbf{F}=-B_x\mathrm{d}x\wedge \mathrm{d}t-B_y\mathrm{d}y\wedge \mathrm{d}t-B_z\mathrm{d}z\wedge \mathrm{d}t+E_x\mathrm{d}y\wedge \mathrm{d}z+E_y\mathrm{d}z\wedge \mathrm{d}x+E_z\mathrm{d}x\wedge \mathrm{d}y}$$

Current 3-form, dual current 1-form

Here, the 3-form J is called the electric current form or current 3-form: $${\displaystyle \mathbf{J}=\rho \mathrm{d}x\wedge \mathrm{d}y\wedge \mathrm{d}z-j_x\mathrm{d}t\wedge \mathrm{d}y\wedge \mathrm{d}z-j_y\mathrm{d}t\wedge \mathrm{d}z\wedge \mathrm{d}x-j_z\mathrm{d}t\wedge \mathrm{d}x\wedge \mathrm{d}y.}$$

That F is a closed form, and the exterior derivative of its Hodge dual is the current 3-form, express Maxwell's equations:

Maxwell's equations

${\displaystyle \mathrm{d}\mathbf{F}=0}$

${\displaystyle \mathrm{d}\star \mathbf{F}=\mathbf{J}}$

Here d denotes the exterior derivative – a natural coordinate- and metric-independent differential operator acting on forms, and the (dual) Hodge star operator ${\displaystyle \star }$ is a linear transformation from the space of 2-forms to the space of (4 − 2)-forms defined by the metric in Minkowski space (in four dimensions even by any metric conformal to this metric). The fields are in natural units where 1/(4πε₀) = 1.

Since d² = 0, the 3-form J satisfies the conservation of current (continuity equation): $${\displaystyle \mathrm{d}\mathbf{J}={\mathrm{d}}^2\star \mathbf{F}=0.}$$ The current 3-form can be integrated over a 3-dimensional space-time region. The physical interpretation of this integral is the charge in that region if it is spacelike, or the amount of charge that flows through a surface in a certain amount of time if that region is a spacelike surface cross a timelike interval. As the exterior derivative is defined on any manifold, the differential form version of the Bianchi identity makes sense for any 4-dimensional manifold, whereas the source equation is defined if the manifold is oriented and has a Lorentz metric. In particular the differential form version of the Maxwell equations are a convenient and intuitive formulation of the Maxwell equations in general relativity.

Note: In much of the literature, the notations ${\displaystyle \mathbf{J}}$ and ${\displaystyle \star \mathbf{J}}$ are switched, so that ${\displaystyle \mathbf{J}}$ is a 1-form called the current and ${\displaystyle \star \mathbf{J}}$ is a 3-form called the dual current.

Linear macroscopic influence of matter

In a linear, macroscopic theory, the influence of matter on the electromagnetic field is described through more general linear transformation in the space of 2-forms. We call $${\displaystyle C:{\mathrm{\Lambda }}^2\ni \mathbf{F}\mapsto \mathbf{G}\in {\mathrm{\Lambda }}^{(4-2)}}$$ the constitutive transformation. The role of this transformation is comparable to the Hodge duality transformation. The Maxwell equations in the presence of matter then become: $${\displaystyle \mathrm{d}\mathbf{F}=0}$$ $${\displaystyle \mathrm{d}\mathbf{G}=\mathbf{J}}$$ where the current 3-form J still satisfies the continuity equation dJ = 0.

When the fields are expressed as linear combinations (of exterior products) of basis forms θⁱ, $${\displaystyle \mathbf{F}=\frac{1}{2}{F}_{pq}{\theta }^p\wedge {\theta }^q.}$$ the constitutive relation takes the form $${\displaystyle G_{pq}=C_{pq}^{mn}{F}_{mn}}$$ where the field coefficient functions and the constitutive coefficients are anticommutative for swapping of each one's indices. In particular, the Hodge star operator that was used in the above case is obtained by taking $${\displaystyle C_{pq}^{mn}=\frac{1}{2}{g}^{ma}{g}^{nb}{\epsilon }_{abpq}\sqrt{-g}}$$ in terms of tensor index notation with respect to a (not necessarily orthonormal) basis $\left\{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1},\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}\right\}$ in a tangent space ${\displaystyle V=T_{p}M}$ and its dual basis ${\displaystyle \{d{x}_1,\dots ,d{x}_n\}}$ in ${\displaystyle V^{\ast }=T_p^{\ast }M}$, having the gram metric matrix $(g_{ij})=\left(⟨\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i},\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}⟩\right)$ and its inverse matrix ${\displaystyle (g^{ij})=(⟨d{x}^i,d{x}^j⟩)}$, and ${\displaystyle {\epsilon }_{abpq}}$ is the Levi-Civita symbol with ${\displaystyle {\epsilon }_{1234}=1}$. Up to scaling, this is the only invariant tensor of this type that can be defined with the metric.

In this formulation, electromagnetism generalises immediately to any 4-dimensional oriented manifold or with small adaptations any manifold.

Alternative metric signature

In the particle physicist's sign convention for the metric signature (+ − − −), the potential 1-form is $${\displaystyle \mathbf{A}=\varphi \mathrm{d}t-A_x\mathrm{d}x-A_y\mathrm{d}y-A_z\mathrm{d}z.}$$

The Faraday curvature 2-form becomes $${\displaystyle \begin{array}{rl}\mathbf{F}\equiv & \frac{1}{2}{F}_{\mu \nu }\mathrm{d}{x}^{\mu }\wedge \mathrm{d}{x}^{\nu }\\ =& E_x\mathrm{d}t\wedge \mathrm{d}x+E_y\mathrm{d}t\wedge \mathrm{d}y+E_z\mathrm{d}t\wedge \mathrm{d}z-B_x\mathrm{d}y\wedge \mathrm{d}z-B_y\mathrm{d}z\wedge \mathrm{d}x-B_z\mathrm{d}x\wedge \mathrm{d}y\end{array}}$$ and the Maxwell tensor becomes $${\displaystyle \star \mathbf{F}=-E_x\mathrm{d}y\wedge \mathrm{d}z-E_y\mathrm{d}z\wedge \mathrm{d}x-E_z\mathrm{d}x\wedge \mathrm{d}y-B_x\mathrm{d}t\wedge \mathrm{d}x-B_y\mathrm{d}t\wedge \mathrm{d}y-B_z\mathrm{d}t\wedge \mathrm{d}z.}$$

The current 3-form J is $${\displaystyle \mathbf{J}=-\rho \mathrm{d}x\wedge \mathrm{d}y\wedge \mathrm{d}z+j_x\mathrm{d}t\wedge \mathrm{d}y\wedge \mathrm{d}z+j_y\mathrm{d}t\wedge \mathrm{d}z\wedge \mathrm{d}x+j_z\mathrm{d}t\wedge \mathrm{d}x\wedge \mathrm{d}y}$$ and the corresponding dual 1-form is $${\displaystyle \star \mathbf{J}=-\rho \mathrm{d}t+j_x\mathrm{d}x+j_y\mathrm{d}y+j_z\mathrm{d}z.}$$

The current norm is now positive and equals $${\displaystyle \mathbf{J}\wedge \star \mathbf{J}=[{\rho }^2+(j_x{)}^2+(j_y{)}^2+(j_z{)}^2]\star (1)}$$ with the canonical volume form ${\displaystyle \star (1)=\mathrm{d}t\wedge \mathrm{d}x\wedge \mathrm{d}y\wedge \mathrm{d}z}$.

Curved spacetime

Main article: Maxwell's equations in curved spacetime

Traditional formulation

Matter and energy generate curvature of spacetime. This is the subject of general relativity. Curvature of spacetime affects electrodynamics. An electromagnetic field having energy and momentum also generates curvature in spacetime. Maxwell's equations in curved spacetime can be obtained by replacing the derivatives in the equations in flat spacetime with covariant derivatives. (Whether this is the appropriate generalization requires separate investigation.) The sourced and source-free equations become (cgs-Gaussian units): $${\displaystyle \frac{4\pi }{c}{j}^{\beta }={\mathrm{\partial }}_{\alpha }{F}^{\alpha \beta }+{{\mathrm{\Gamma }}^{\alpha }}_{\mu \alpha }{F}^{\mu \beta }+{{\mathrm{\Gamma }}^{\beta }}_{\mu \alpha }{F}^{\alpha \mu }\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\mathrm{\nabla }}_{\alpha }{F}^{\alpha \beta }\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{F^{\alpha \beta }}_{;\alpha }}$$ and $${\displaystyle 0={\mathrm{\partial }}_{\gamma }{F}_{\alpha \beta }+{\mathrm{\partial }}_{\beta }{F}_{\gamma \alpha }+{\mathrm{\partial }}_{\alpha }{F}_{\beta \gamma }={\mathrm{\nabla }}_{\gamma }{F}_{\alpha \beta }+{\mathrm{\nabla }}_{\beta }{F}_{\gamma \alpha }+{\mathrm{\nabla }}_{\alpha }{F}_{\beta \gamma }.}$$

Here, $${\displaystyle {{\mathrm{\Gamma }}^{\alpha }}_{\mu \beta }}$$ is a Christoffel symbol that characterizes the curvature of spacetime and ∇_α is the covariant derivative.

Formulation in terms of differential forms

The formulation of the Maxwell equations in terms of differential forms can be used without change in general relativity. The equivalence of the more traditional general relativistic formulation using the covariant derivative with the differential form formulation can be seen as follows. Choose local coordinates x^α that gives a basis of 1-forms dx^α in every point of the open set where the coordinates are defined. Using this basis and cgs-Gaussian units we define

• The antisymmetric field tensor F_αβ, corresponding to the field 2-form F $${\displaystyle \mathbf{F}=\frac{1}{2}{F}_{\alpha \beta }\mathrm{d}{x}^{\alpha }\wedge \mathrm{d}{x}^{\beta }.}$$
• The current-vector infinitesimal 3-form J $${\displaystyle \mathbf{J}=\frac{4\pi }{c}\left(\frac{1}{6}{j}^{\alpha }\sqrt{-g}{\epsilon }_{\alpha \beta \gamma \delta }\mathrm{d}{x}^{\beta }\wedge \mathrm{d}{x}^{\gamma }\wedge \mathrm{d}{x}^{\delta }.\right)}$$

The epsilon tensor contracted with the differential 3-form produces 6 times the number of terms required.

Here g is as usual the determinant of the matrix representing the metric tensor, g_αβ. A small computation that uses the symmetry of the Christoffel symbols (i.e., the torsion-freeness of the Levi-Civita connection) and the covariant constantness of the Hodge star operator then shows that in this coordinate neighborhood we have:

• the Bianchi identity $${\displaystyle \mathrm{d}\mathbf{F}=2({\mathrm{\partial }}_{\gamma }{F}_{\alpha \beta }+{\mathrm{\partial }}_{\beta }{F}_{\gamma \alpha }+{\mathrm{\partial }}_{\alpha }{F}_{\beta \gamma })\mathrm{d}{x}^{\alpha }\wedge \mathrm{d}{x}^{\beta }\wedge \mathrm{d}{x}^{\gamma }=0,}$$
• the source equation $${\displaystyle \mathrm{d}\star \mathbf{F}=\frac{1}{6}{F^{\alpha \beta }}_{;\alpha }\sqrt{-g}{\epsilon }_{\beta \gamma \delta \alpha }\mathrm{d}{x}^{\gamma }\wedge \mathrm{d}{x}^{\delta }\wedge \mathrm{d}{x}^{\alpha }=\mathbf{J},}$$
• the continuity equation $${\displaystyle \mathrm{d}\mathbf{J}=\frac{4\pi }{c}{j^{\alpha }}_{;\alpha }\sqrt{-g}{\epsilon }_{\alpha \beta \gamma \delta }\mathrm{d}{x}^{\alpha }\wedge \mathrm{d}{x}^{\beta }\wedge \mathrm{d}{x}^{\gamma }\wedge \mathrm{d}{x}^{\delta }=0.}$$

Classical electrodynamics as the curvature of a line bundle

An elegant and intuitive way to formulate Maxwell's equations is to use complex line bundles or a principal ${\displaystyle \mathrm{U}(1)}$-bundle, on the fibers of which U(1) acts regularly. The principal U(1)-connection ∇ on the line bundle has a curvature F = ∇², which is a two-form that automatically satisfies dF = 0 and can be interpreted as a field strength. If the line bundle is trivial with flat reference connection d we can write ∇ = d + A and F = dA with A the 1-form composed of the electric potential and the magnetic vector potential.

In quantum mechanics, the connection itself is used to define the dynamics of the system. This formulation allows a natural description of the Aharonov–Bohm effect. In this experiment, a static magnetic field runs through a long magnetic wire (e.g., an iron wire magnetized longitudinally). Outside of this wire the magnetic induction is zero, in contrast to the vector potential, which essentially depends on the magnetic flux through the cross-section of the wire and does not vanish outside. Since there is no electric field either, the Maxwell tensor F = 0 throughout the space-time region outside the tube, during the experiment. This means by definition that the connection ∇ is flat there.

In mentioned Aharonov–Bohm effect, however, the connection depends on the magnetic field through the tube since the holonomy along a non-contractible curve encircling the tube is the magnetic flux through the tube in the proper units. This can be detected quantum-mechanically with a double-slit electron diffraction experiment on an electron wave traveling around the tube. The holonomy corresponds to an extra phase shift, which leads to a shift in the diffraction pattern.

Discussion and other approaches

Following are the reasons for using each of such formulations.

Potential formulation

In advanced classical mechanics it is often useful, and in quantum mechanics frequently essential, to express Maxwell's equations in a potential formulation involving the electric potential (also called scalar potential) φ, and the magnetic potential (a vector potential) A. For example, the analysis of radio antennas makes full use of Maxwell's vector and scalar potentials to separate the variables, a common technique used in formulating the solutions of differential equations. The potentials can be introduced by using the Poincaré lemma on the homogeneous equations to solve them in a universal way (this assumes that we consider a topologically simple, e.g. contractible space). The potentials are defined as in the table above. Alternatively, these equations define E and B in terms of the electric and magnetic potentials that then satisfy the homogeneous equations for E and B as identities. Substitution gives the non-homogeneous Maxwell equations in potential form.

Many different choices of A and φ are consistent with given observable electric and magnetic fields E and B, so the potentials seem to contain more, (classically) unobservable information. The non uniqueness of the potentials is well understood, however. For every scalar function of position and time λ(x, t), the potentials can be changed by a gauge transformation as $${\displaystyle {\phi }^{\prime }=\phi -\frac{\mathrm{\partial }\lambda }{\mathrm{\partial }t},{\mathbf{A}}^{\prime }=\mathbf{A}+\mathrm{\nabla }\lambda }$$ without changing the electric and magnetic field. Two pairs of gauge transformed potentials (φ, A) and (φ′, A′) are called gauge equivalent, and the freedom to select any pair of potentials in its gauge equivalence class is called gauge freedom. Again by the Poincaré lemma (and under its assumptions), gauge freedom is the only source of indeterminacy, so the field formulation is equivalent to the potential formulation if we consider the potential equations as equations for gauge equivalence classes.

The potential equations can be simplified using a procedure called gauge fixing. Since the potentials are only defined up to gauge equivalence, we are free to impose additional equations on the potentials, as long as for every pair of potentials there is a gauge equivalent pair that satisfies the additional equations (i.e. if the gauge fixing equations define a slice to the gauge action). The gauge-fixed potentials still have a gauge freedom under all gauge transformations that leave the gauge fixing equations invariant. Inspection of the potential equations suggests two natural choices. In the Coulomb gauge, we impose ∇ ⋅ A = 0, which is mostly used in the case of magneto statics when we can neglect the c⁻²∂²A/∂t² term. In the Lorenz gauge (named after the Dane Ludvig Lorenz), we impose $${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}+\frac{1}{c^2}\frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}=0.}$$ The Lorenz gauge condition has the advantage of being Lorentz invariant and leading to Lorentz-invariant equations for the potentials.

Manifestly covariant (tensor) approach

Main articles: Classical electromagnetism and special relativity and Covariant formulation of classical electromagnetism

Maxwell's equations are exactly consistent with special relativity—i.e., if they are valid in one inertial reference frame, then they are automatically valid in every other inertial reference frame. In fact, Maxwell's equations were crucial in the historical development of special relativity. However, in the usual formulation of Maxwell's equations, their consistency with special relativity is not obvious; it can only be proven by a laborious calculation.

For example, consider a conductor moving in the field of a magnet. In the frame of the magnet, that conductor experiences a magnetic force. But in the frame of a conductor moving relative to the magnet, the conductor experiences a force due to an electric field. The motion is exactly consistent in these two different reference frames, but it mathematically arises in quite different ways.

For this reason and others, it is often useful to rewrite Maxwell's equations in a way that is "manifestly covariant"—i.e. obviously consistent with special relativity, even with just a glance at the equations—using covariant and contravariant four-vectors and tensors. This can be done using the EM tensor F, or the 4-potential A, with the 4-current J.

Differential forms approach

Gauss's law for magnetism and the Faraday–Maxwell law can be grouped together since the equations are homogeneous, and be seen as geometric identities expressing the field F (a 2-form), which can be derived from the 4-potential A. Gauss's law for electricity and the Ampere–Maxwell law could be seen as the dynamical equations of motion of the fields, obtained via the Lagrangian principle of least action, from the "interaction term" AJ (introduced through gauge covariant derivatives), coupling the field to matter. For the field formulation of Maxwell's equations in terms of a principle of extremal action, see electromagnetic tensor.

Often, the time derivative in the Faraday–Maxwell equation motivates calling this equation "dynamical", which is somewhat misleading in the sense of the preceding analysis. This is rather an artifact of breaking relativistic covariance by choosing a preferred time direction. To have physical degrees of freedom propagated by these field equations, one must include a kinetic term F ⋆F for A, and take into account the non-physical degrees of freedom that can be removed by gauge transformation A ↦ A − dα. See also gauge fixing and Faddeev–Popov ghosts.

Geometric calculus approach

This formulation uses the algebra that spacetime generates through the introduction of a distributive, associative (but not commutative) product called the geometric product. Elements and operations of the algebra can generally be associated with geometric meaning. The members of the algebra may be decomposed by grade (as in the formalism of differential forms) and the (geometric) product of a vector with a k-vector decomposes into a (k − 1)-vector and a (k + 1)-vector. The (k − 1)-vector component can be identified with the inner product and the (k + 1)-vector component with the outer product. It is of algebraic convenience that the geometric product is invertible, while the inner and outer products are not. As such, powerful techniques such as Green's functions can be used. The derivatives that appear in Maxwell's equations are vectors and electromagnetic fields are represented by the Faraday bivector F. This formulation is as general as that of differential forms for manifolds with a metric tensor, as then these are naturally identified with r-forms and there are corresponding operations. Maxwell's equations reduce to one equation in this formalism. This equation can be separated into parts as is done above for comparative reasons.

Forest restoration

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Forest_restoration 

In the 1980s, conservation organizations warned that, once destroyed, tropical forests could never be restored. Thirty years of restoration research now challenge this: a) This site in Doi Suthep-Pui National Park, N. Thailand was deforested, over-cultivated and then burnt. The black tree stump was one of the original forest trees. Local people teamed up with scientists to repair their watershed.

b) Fire prevention, nurturing natural regeneration and planting framework tree species resulted in trees growing above the weed canopy within a year.

c) After 12 years, the restored forest overwhelmed the black tree stump.

Forest restoration is defined as "actions to re-instate ecological processes, which accelerate recovery of forest structure, ecological functioning and biodiversity levels towards those typical of climax forest", i.e. the end-stage of natural forest succession. Climax forests are relatively stable ecosystems that have developed the maximum biomass, structural complexity and species diversity that are possible within the limits imposed by climate and soil and without continued disturbance from humans (more explanation here). Climax forest is therefore the target ecosystem, which defines the ultimate aim of forest restoration. Since climate is a major factor that determines climax forest composition, global climate change may result in changing restoration aims. Additionally, the potential impacts of climate change on restoration goals must be taken into account, as changes in temperature and precipitation patterns may alter the composition and distribution of climax forests.

Forest restoration is a specialized form of reforestation, but it differs from conventional tree plantations in that its primary goals are biodiversity recovery and environmental protection.

Forest and landscape restoration (FLR) is defined as a process that aims to regain ecological functionality and enhance human well-being in deforested or degraded landscapes. FLR has been developed as a response to the growing degradation and loss of forest and land, which resulted in declined biodiversity and ecosystem services. Effective FLR will support the achievement of the Sustainable Development Goals. The United Nations Decade on Ecosystem Restoration (2021–2030) provides the opportunity to restore hundreds of millions of hectares of degraded forests and other ecosystems. Successful ecosystem restoration requires a fundamental understanding of the ecological characteristics of the component species, together with knowledge of how they assemble, interact and function as communities

Scope

Forest restoration may include simply protecting remnant vegetation (fire prevention, cattle exclusion etc.) or more active interventions to accelerate natural regeneration, as well as tree planting and/or sowing seeds (direct seeding) of species characteristic of the target ecosystem. Tree species planted (or encouraged to establish) are those that are typical of, or provide a critical ecological function in, the target ecosystem. However, wherever people live in or near restoration sites, restoration projects often include economic species amongst the planted trees, to yield subsistence or cash-generating products.

Forest restoration is an inclusive process, which depends on collaboration among a wide range of stakeholders including local communities, government officials, non-government organizations, scientists and funding agencies. Its ecological success is measured in terms of increased biological diversity, biomass, primary productivity, soil organic matter and water-holding capacity, as well as the return of rare and keystone species, characteristic of the target ecosystem. However, according to FAO, restoration activities face economic barriers ranging from a lack of large-scale funding available on behalf of governments to the limited resources and technical capacity of smallholders.

Economic indices of success include the value of forest products and ecological services generated (e.g. watershed protection, carbon storage etc.), which ultimately contribute towards poverty reduction. Payments for such ecological services (PES) and forest products can provide strong incentives for local people to implement restoration projects. Active restoration has been shown to accelerate the carbon recovery of human-modified tropical forests by as much as 50%.

According to FAO's The State of the World's Forests 2020, large-scale forest restoration is needed to meet the Sustainable Development Goals and to prevent, halt and reverse the loss of biodiversity. While 61 countries have, together, pledged to restore 170 million hectares of degraded forest lands under the Bonn Challenge, progress to date is slow. Forest restoration, when implemented appropriately, helps restore habitats and ecosystems, create jobs and income and is an effective nature-based solution to climate change. Moreover, according to FAO, forest and landscape restoration yields many benefits for the climate, including greenhouse gas emissions sequestration and reduction. The United Nations Decade on Ecosystem Restoration 2021–2030, announced in March 2019, aims to accelerate ecosystem restoration action worldwide.

Opportunities for forest restoration

Demonstration forest restoration plot, SUNY-ESF, Syracuse, New York

Forest restoration is appropriate wherever biodiversity recovery is one of the main goals of reforestation, such as for wildlife conservation, environmental protection, eco-tourism or to supply a wide variety of forest products to local communities. Forests can be restored in a wide range of circumstances, but degraded sites within protected areas are a high priority, especially where some climax forest remains as a seed source within the landscape. Even in protected areas, there are often large deforested sites: logged over areas or sites formerly cleared for agriculture. If protected areas are to act as Earth's last wildlife refuges, restoration of such areas will be needed.

Many restoration projects are now being implemented under the umbrella of "forest landscape restoration" (FLR), defined as a "planned process to regain ecological integrity and enhance human well-being in deforested or degraded landscapes". FLR recognizes that forest restoration has social and economic functions. It aims to achieve the best possible compromise between meeting both conservation goals and the needs of rural communities. As human pressure on landscapes increases, forest restoration will most commonly be practiced within a mosaic of other forms of forest management, to meet the economic needs of local people.

A recent focal area for forest restoration efforts is within the urban context, where both people and biodiversity will benefit, however this context presents unique challenges.

Natural regeneration

Tree planting is not always essential to restore forest ecosystems. A lot can be achieved by studying how forests regenerate naturally, identifying the factors that limit regeneration and devising methods to overcome them. These can include weeding and adding fertilizer around natural tree seedlings, preventing fire, removing cattle and so on. This is "accelerated" or "assisted" natural regeneration. It is simple and cost-effective, but it can only operate on trees that are already present, mostly light-loving pioneer species. Such tree species are not usually those that comprise climax forests, but they can foster recolonization of the site by shade-tolerant climax forest tree species, via natural seed dispersal from remnant forest. Because this is a slow process, biodiversity recovery can usually be accelerated by planting some climax forest tree species, especially large-seeded, poorly dispersed species. It is not feasible to plant all the tree species that may have formerly grown in the original primary forest and it is usually unnecessary to do so, if the framework species method can be used.

In some exceptional cases, particularly some Alaskan boreal forests, the long-term recovery from wildfires could offset the carbon emitted during the fires due to a change in tree species if the trees persist, prove to become part of resilient biomes and are about as numerous as the former forests'.

Protecting regeneration areas from browsing animals

Forest in the process of restoration face many challenges, such as seed and nutrient availability, but are notable susceptible to browsing animals. Although browsing animals are necessary in maintaining the understory of forests, they can easily over-graze a freshly replanted swath of forest, where young samplings are easily accessible. Over-grazing is particularly problematic in this case as the samplings and other young plants may be damaged beyond the point of recovery, resulting in a decrease in biodiversity. Care must be taken to use "deer fencing" to protect the regeneration area, or where not financially possible, to plant trees which prioritize structural growth and recovery.

Post-fire regeneration

In large parts of the world, forest fires cover a heavy toll on forests. That can be because of provoked deforestation in order to substitute forests by crop areas, or in dry areas, because of wild fires occurring naturally or intentionally. A whole section of forest landscape restoration in linked to this particular problem, as in many cases, the net loss of ecosystem value is very high and can open the drop to an accelerated further degradation of the soil conditions through erosion and desertification. This indeed has dire consequences on both the quality of the habitats and their related fauna. Nevertheless, in some specific cases, wild fires do actually allow to increase the biodiversity index of the burnt area, in which case the Forest Restoration Strategies tend to look for a different land-use.

Forest restoration projects

A study finds that almost 300 million people live on tropical forest restoration opportunity land in the Global South, constituting a large share of low-income countries' populations, and argues for prioritized inclusion of "local communities" in forest restoration projects. Project Drawdown lists the restoration of tropical forests as one of the most important solutions for climate change mitigation due to its extraordinary potential to sequestrate carbon and recommends that "local communities need to have a stake in what is growing, if restoration is to sustain." A recent FAO publication reports that Indigenous Peoples are among those facing the greatest risk to their well-being and livelihoods from the effects of climate change, and therefore must be centred in forest restoration and conservation.

Ashland Forest Resiliency Stewardship Project

The Ashland Forest Resiliency Stewardship Project (AFR) is a decade long, science-based project launched in 2010 with the intent of reducing severe wildfire risk, but also protecting water quality, old-growth forest, wildlife, people, property, and the overall quality of life within the Ashland watershed. The primary stakeholders in this cooperative restoration effort are the U.S. Forest Service, the City of Ashland, Lomakatsi Restoration Project, and The Nature Conservancy. The project was launched with initial funding from the Economic Recovery stimulus, and received funding from the Forest Service Hazardous Fuels program and the Joint Chiefs Landscape Restoration Partnerships program to back the project through 2016.

Located in the dry forests of southern Oregon, the threat of wildfire is a reality for land managers and property owners alike. The boundaries of the city of Ashland intersect with the surrounding forest in what is referred to as the wildland–urban interface (WUI). Historically, the forests of this region experienced a relatively frequent fire return interval, which prevented buildup of heavy fuel loads. A century of fire exclusion and suppression on federal lands in the Pacific Northwest has led to increased forest density and fuel loads, and thus a more persistent threat of devastating wildfire.

The AFR project has implemented restoration techniques and prescriptions that aim to replicate the process of ecological succession in dry, mixed-conifer forests of the Pacific Northwest. The approach involves a combination of fuels reduction, thinning small-diameter trees, and carrying out prescribed burns. Priority is given to maintaining ecological function and complexity by retaining the largest and oldest trees, preserving wildlife habitat and riparian areas, and protecting erodible soils and maintaining slope stability.

Since its inception in 2010, the AFR project has provided educational experiences to thousands of students and has benefitted the local community by creating jobs and providing workforce training. About 13,000 acres treated in the AFR project was in maintenance status as of February 2022, and Oregon's Landscape Resiliency Program, established through Senate Bill 762, is funding brush cutting and low-intensity burns to ecologically benefit a fire-adapted forest.

Accelerating forest regeneration with Agricultural Waste

In 1998, Costa Rican initiatives were set to regenerate deforested areas, formerly used as cattle pasture. This land was compacted and the soil was depleted, making natural regeneration more difficult. As a partnership with agricultural waste disposals, approximately 12 000 Mg of orange peels and pulp were applied to a 3 hectare segment of the former pastures. This addition of biomass to the soil allowed for a 176% increase in woody plant growth, increased species richness, tripled tree evenness (measured through the Shannon Index), and significantly elevated soil nutrient levels, measured at 2 and 16 years following the application (Truer et al. 2018). A significant increase in canopy closure was also observed using hemispheric topography, further suggesting that agricultural waste may play a larger role in future forest restoration.

Forest landscape restoration

Forest landscape restoration (FLR) is defined as "a planned process to regain ecological integrity and enhance human well-being in deforested or degraded landscapes". It comprises tools and procedures to integrate site-level forest restoration actions with desirable landscape-level objectives, which are decided upon via various participatory mechanisms among stakeholders. The concept has grown out of collaboration among some of the world's major international conservation organizations including the International Union for Conservation of Nature (IUCN), the World Wide Fund for Nature (WWF), the World Resources Institute and the International Tropical Timber Organization (ITTO).

Aims

The concept of FLR was conceived to bring about compromises between meeting the needs of both humans and wildlife, by restoring a range of forest functions at the landscape level. It includes actions to strengthen the resilience and ecological integrity of landscapes and thereby keep future management options open. The participation of local communities is central to the concept, because they play a critical role in shaping the landscape and gain significant benefits from restored forest resources. Therefore, FLR activities are inclusive and participatory.

Desirable outcomes

The desirable outcomes of an FLR program usually comprise a combination of the following, depending on local needs and aspirations:

• identification of the root causes of forest degradation and prevention of further deforestation,
• positive engagement of people in the planning of forest restoration, resolution of land-use conflicts and agreement on benefit-sharing systems,
• compromises over land-use trade-offs that are acceptable to the majority of stakeholders,
• a repository of biological diversity of both local and global value,
• delivery of a range of utilitarian benefits to local communities including:
  • a reliable supply of clean water,
  • environmental protection particularly watershed services (e.g. reduced soil erosion, lower landslide risk, flood/drought mitigation etc.),
  • a sustainable supply of a diverse range of forest products including foods, medicines, firewood etc.,
  • monetary income from various sources e.g. ecotourism, carbon trading via the REDD+ mechanism and from payments for other environmental services (PES)

Activities

FLR combines several existing principles and techniques of development, conservation and natural resource management, such as landscape character assessment, participatory rural appraisal, adaptive management etc. within a clear and consistent evaluation and learning framework. An FLR program may comprise various forestry practices on different sites within the landscape, depending on local environmental and socioeconomic factors. These may include protection and management of secondary and degraded primary forests, standard forest restoration techniques such as "assisted" or "accelerated" natural regeneration (ANR) and the planting of framework tree species to restore degraded areas, as well as conventional tree plantations and agroforestry systems to meet more immediate monetary needs.

The IUCN hosts the Global Partnership on Forest Landscape Restoration, which co-ordinates development of the concept around the world.

In 2014, the Food and Agricultural Organization of the United Nations established the Forest and Landscape Restoration Mechanism. The Mechanism supports countries to implement FLR as a contribution to achieving the Bonn Challenge—the restoration of 150 million hectare of deforested and degraded lands by 2020—and the Convention on Biological Diversity Aichi Biodiversity Targets—related to ecosystem conservation and restoration.

In partnership with the Global Mechanism of the United Nations Convention to Combat Desertification, FAO released two discussion papers on sustainable financing for FLR in 2015. Sustainable Financing for Forest and Landscape Restoration: The Role of Public Policy Makers provides recommendations and examples of FLR financing for countries. Sustainable Financing for Forest and Landscape Restoration – Opportunities, challenges and the way forward provides an overview of funding sources and financial instruments available for FLR activities.

Financing

To finance the planning and implementation of forest and landscape restoration (FLR) activities, the Food and Agriculture Organization of the United Nations (FAO) has identified diverse financial mechanisms that are tailored to different stages of the FLR process and cover the transaction and the scaling-up of enterprises for sustainable restoration. Several options are available to finance restoration. To meet the unique demands of individual FLR projects, it is critical to identify the best landscape financing strategy. Financial options that generate diverse incentives for local actors may either be for-profit mechanisms, such as debt or loans, or not-for-profit mechanisms, including grants, fiscal policies, or expenses by the public sector. According to FAO, bridging the gap between smallholders and investors, coordinating investment, promoting local ownership of FLR financing strategies, and developing bankable projects and blended financial mechanisms generate positive outcomes for FLR impact at scale.