Alpha effect ( $E = α \overline{B}$ )

The alpha effect is the part of a mean-field electromotive force linear in the mean magnetic field itself,

E = α \overline{B} + \dots

. The alpha tensor can be anisotropic; it need not be a scalar multiple of the identity. Its curl can couple transverse mean-field components and produce an alpha-squared dynamo.

Table of contents
- Alpha-squared dynamo Alpha effect
  - Anisotropic alpha-squared dynamo Alpha-squared dynamo
    - Uniaxial alpha dynamo threshold Anisotropic alpha-squared dynamo
- Alpha tensor Alpha effect

Alpha-squared dynamo ( $α^{2}$ dynamo)

 0  0

An alpha-squared dynamo generates a mean magnetic field using an alpha effect without requiring an additional large-scale shear coupling. For

α = α_{0} diag (1, 1, 0)

and a transverse mean field varying as

e^{i Kz}

, the growth rates are

- η K^{2} \pm α_{0} K

. A nonzero

α_{0}

therefore allows growth for sufficiently long wavelengths, provided the domain and scale separation permit them.

Anisotropic alpha-squared dynamo ( $\partial_{t} B = \nabla \times (α B) + η \nabla^{2} B$ )

 0  0

An alpha-squared dynamo may use a tensorial alpha effect rather than an isotropic coefficient. For constant alpha tensor

α

and magnetic diffusivity

η > 0

, a steady nonzero Fourier mode with wavevector

K

satisfies

η K^{2} B = i K \times (α B)

and

K \cdot B = 0

. Anisotropy can change both the critical alpha magnitude and the preferred wavevector direction.

Uniaxial alpha dynamo threshold ( $γ^{2} = (q^{2} + m^{2})^{2} / (m^{2} + δ q^{2})$ )

 0  0

For

α = diag (α_{0}, α_{0}, α_{1})

, set

γ = α_{0} / η

δ = α_{1} / α_{0}

, and

q^{2} = k^{2} + l^{2}

. A steady nonzero Fourier mode obeys the displayed condition when the denominator is nonzero. At fixed

q > 0

and

0 < δ < 1

, minimizing over

m^{2} \geq 0

gives

m_{*}^{2} = (1 - 2 δ) q^{2}

and

γ_{m i n}^{2} = 4 (1 - δ) q^{2}

for

δ < 1/2

; for

δ \geq 1/2

it gives

m_{*}^{2} = 0

and

γ_{m i n}^{2} = q^{2} / δ

. This is a boundary-constrained threshold of an anisotropic alpha-squared dynamo.

Alpha tensor ( $α_{ij}$ )

 0  0

The alpha tensor is the linear map from a uniform test magnetic field to the corresponding mean-field electromotive force. Its components satisfy

E_{i} = α_{ij} B_{0 j}

. Anisotropic helical flows can produce a real symmetric rank-two response, such as

α_{0} diag (1, 1, 0)

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The term "Alpha effect" can refer to different concepts depending on the context. Here are a few predominant uses of the term: 1. **Finance and Investments**: In finance, the Alpha effect relates to the performance of an investment relative to a benchmark index, usually in the context of active portfolio management. Alpha is a measure of the excess return of an investment compared to a market index.

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Alpha effect ( $E = α \overline{B}$ )

Alpha-squared dynamo ( $α^{2}$ dynamo)

Anisotropic alpha-squared dynamo ( $\partial_{t} B = \nabla \times (α B) + η \nabla^{2} B$ )

Uniaxial alpha dynamo threshold ( $γ^{2} = (q^{2} + m^{2})^{2} / (m^{2} + δ q^{2})$ )

Alpha tensor ( $α_{ij}$ )

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