Número de Elsasser
# Número de Elsasser

El número de Elsasser, Λ, es un número adimensional en magnetohidrodinámica que representa la relación entre las fuerzas magnéticas y la fuerza de Coriolis.

## Simbología

Simbología :

| Símbolo | Nombre | Unidad |
| - | - | - |
| $\Lambda _{i}$ | Número de Elsasser impuesto | |
| $\Lambda _{d}$ | Número de Elsasser dinámico | |
| $\mathrm {Ek}$ | Número de Ekman | |
| $\mathrm {P_{m}}$ | Número de Prandtl magnético | |
| $\mathrm {R_{m}}$ | Número de Reynolds magnético | |
| | Dimensiones | |
| $\ell$ | | m |
| $L$ | Longitud | m |
| $d$ | Dimensión de sección transversal | m |
| $t$ | Tiempo | s |
| | Magnético | |
| $B$ | Densidad de flujo magnético | T |
| $H$ | Fuerza de campo magnético | A / m |
| $\rho _{m}$ | Densidad de energía magnética | J / $m^3$ |
| $q$ | Carga | C |
| $\eta$ | Difusividad magnética | $m^2$ / s |
| $\mu _{0}$ | Permeabilidad en el vacío | H / m |
| $\theta$ | Ángulo entre velocidad ($u$) y campo magnético ($B$) | |
| | Materia | |
| $m$ | Masa | kg |
| ${\dot {m}}$ | Flujo másico | kg / s |
| $u$ | Velocidad | m / s |
| $\rho$ | Densidad | kg / $m^3$ |
| $\nu$ | Viscosidad cinemática | $m^2$ / s |
| | Planeta | |
| $\phi$ | Latitud | |
| $\Omega$ | Velocidad de rotación del cuerpo | m / s |

## Descripción

Se define como:

$\Lambda ={\frac {\text{Fuerza magnética}}{\text{Fuerza de Coriolis}}}$

$\Lambda _{i}={\frac {q\ u\ B\sin \theta }{{\dot {m}}\ L\ (2\Omega \sin \phi )}}$

### Número de Elsasser Impuesto (Bajo $\mathrm {R_{m}}$)

Deducción :

| | 1 | 2 | 3 | 4 | 5 | 6 |
| - | - | - | - | - | - | - |
| Ecuaciones | $\Lambda _{i}={\frac {q\ u\ B}{{\dot {m}}\ L\ (2\Omega )}}$ | ${\dot {m}}={\frac {m}{t}}$ | $u={\frac {L}{t}}$ | $\rho ={\frac {m}{d^{2}L}}$ | $\eta ={\frac {d^{2}}{t}}$ | $\mu _{0}=B{\Bigl{\frac {L\ t}{q}}{\Bigr )}$ |
| Sustituyendo | $\Lambda _{i}={\frac {q\ (L/t)\ B}{(m/t)\ L\ (2\Omega )}}$ | $\Lambda _{i}={\frac {q\ (L/t)\ B}{(m/t)\ L\ (2\Omega )}}$ | $\Lambda _{i}={\frac {q\ (L/t)\ B}{(m/t)\ L\ (2\Omega )}}$ | | | |
| Multiplicando ${\Bigl{\frac {B\ d^{2}}{B\ d^{2}}}{\Bigr )}$ | $\Lambda _{i}={\frac {q\ (L/t)\ B}{(m/t)\ L\ (2\Omega )}}{\Bigl{\frac {B\ d^{2}}{B\ d^{2}}}{\Bigr )}$ | $\Lambda _{i}={\frac {q\ (L/t)\ B}{(m/t)\ L\ (2\Omega )}}{\Bigl{\frac {B\ d^{2}}{B\ d^{2}}}{\Bigr )}$ | $\Lambda _{i}={\frac {q\ (L/t)\ B}{(m/t)\ L\ (2\Omega )}}{\Bigl{\frac {B\ d^{2}}{B\ d^{2}}}{\Bigr )}$ | | | |
| Ordenando | $\Lambda _{i}={\frac {B^{2}}{2\ [m/(d^{2}L)]\ [d^{2}/t]\ [B\ (L\ t/q)]\ \Omega }}$ | $\Lambda _{i}={\frac {B^{2}}{2\ [m/(d^{2}L)]\ [d^{2}/t]\ [B\ (L\ t/q)]\ \Omega }}$ | $\Lambda _{i}={\frac {B^{2}}{2\ [m/(d^{2}L)]\ [d^{2}/t]\ [B\ (L\ t/q)]\ \Omega }}$ | | | |
| Sustituyendo | $\Lambda _{i}={\frac {B^{2}}{2\ \rho \ \eta \ \mu _{0}\ \Omega }}$ | $\Lambda _{i}={\frac {B^{2}}{2\ \rho \ \eta \ \mu _{0}\ \Omega }}$ | $\Lambda _{i}={\frac {B^{2}}{2\ \rho \ \eta \ \mu _{0}\ \Omega }}$ | $\Lambda _{i}={\frac {B^{2}}{2\ \rho \ \eta \ \mu _{0}\ \Omega }}$ | $\Lambda _{i}={\frac {B^{2}}{2\ \rho \ \eta \ \mu _{0}\ \Omega }}$ | $\Lambda _{i}={\frac {B^{2}}{2\ \rho \ \eta \ \mu _{0}\ \Omega }}$ |

$\Lambda _{i}={\frac {B^{2}}{2\ \rho \ \eta \ \mu _{0}\ \Omega }}$

Deducción :

| | 1 | 2 | 3 | 4 |
| - | - | - | - | - |
| Ecuaciones | $\Lambda _{i}={\frac {B^{2}}{2\ \rho \ \eta \ \mu _{0}\ \Omega }}$ | $\rho _{m}={\frac {B^{2}}{\mu _{0}}}$ | $\mathrm {P_{m}} ={\frac {\nu }{\eta }}$ | $\mathrm {Ek} ={\frac {\nu }{L^{2}\ (2\Omega )}}$ |
| Multiplicando ${\Bigl{\frac {\nu ^{2}L^{2}}{\nu ^{2}L^{2}}}{\Bigr )}$ | $\Lambda _{i}={\frac {B^{2}}{2\ \rho \ \eta \ \mu _{0}\ \Omega }}{\Bigl{\frac {\nu ^{2}L^{2}}{\nu ^{2}L^{2}}}{\Bigr )}$ | | | |
| Ordenando | $\Lambda _{i}={\Bigl{\frac {L^{2}}{\nu ^{2}\ \rho }}{\Bigr )}{\Bigl{\frac {B^{2}}{\mu _{0}}}{\Bigr )}{\Bigl{\frac {\nu }{\eta }}{\Bigr )}{\Bigl{\frac {\nu }{L^{2}(2\ \Omega )}}{\Bigr )}$ | | | |
| Sustituyendo | $\Lambda _{i}={\Bigl{\frac {L^{2}}{\nu ^{2}\ \rho }}{\Bigr )}\rho _{m}\ \mathrm {P_{m}} \ \mathrm {Ek}$ | $\Lambda _{i}={\Bigl{\frac {L^{2}}{\nu ^{2}\ \rho }}{\Bigr )}\rho _{m}\ \mathrm {P_{m}} \ \mathrm {Ek}$ | $\Lambda _{i}={\Bigl{\frac {L^{2}}{\nu ^{2}\ \rho }}{\Bigr )}\rho _{m}\ \mathrm {P_{m}} \ \mathrm {Ek}$ | $\Lambda _{i}={\Bigl{\frac {L^{2}}{\nu ^{2}\ \rho }}{\Bigr )}\rho _{m}\ \mathrm {P_{m}} \ \mathrm {Ek}$ |
| Simplificando | $\Lambda _{i}={\Bigl{\frac {L}{\nu }}{\Bigr )}^{2}{\Bigl{\frac {\rho _{m}}{\rho }}{\Bigr )}\mathrm {P_{m}} \ \mathrm {Ek}$ | $\Lambda _{i}={\Bigl{\frac {L}{\nu }}{\Bigr )}^{2}{\Bigl{\frac {\rho _{m}}{\rho }}{\Bigr )}\mathrm {P_{m}} \ \mathrm {Ek}$ | $\Lambda _{i}={\Bigl{\frac {L}{\nu }}{\Bigr )}^{2}{\Bigl{\frac {\rho _{m}}{\rho }}{\Bigr )}\mathrm {P_{m}} \ \mathrm {Ek}$ | $\Lambda _{i}={\Bigl{\frac {L}{\nu }}{\Bigr )}^{2}{\Bigl{\frac {\rho _{m}}{\rho }}{\Bigr )}\mathrm {P_{m}} \ \mathrm {Ek}$ |

$\Lambda _{i}={\Bigl{\frac {L}{\nu }}{\Bigr )}^{2}{\Bigl{\frac {\rho _{m}}{\rho }}{\Bigr )}\mathrm {P_{m}} \ \mathrm {Ek}$

### Número de Elsasser dinámico (Alto $\mathrm {R_{m}}$)

Deducción :

| | 1 | 2 | 3 | 4 |
| - | - | - | - | - |
| Ecuaciones | $\Lambda _{d}={\frac {B^{2}}{2\ \rho \ \eta \ \mu _{0}\ \Omega }}$ | $\eta ={\frac {d^{2}}{t}}$ | $u={\frac {L}{t}}$ | $\ell ={\frac {d^{2}}{L}}$ |
| Sustituyendo | $\Lambda _{d}={\frac {B^{2}}{2\ \rho \ (d^{2}/t)\ \mu _{0}\ \Omega }}$ | $\Lambda _{d}={\frac {B^{2}}{2\ \rho \ (d^{2}/t)\ \mu _{0}\ \Omega }}$ | | |
| Multiplicando ${\Bigl{\frac {L}{L}}{\Bigr )}$ | $\Lambda _{d}={\frac {B^{2}}{2\ \rho \ (d^{2}/t)\ \mu _{0}\ \Omega }}{\Bigl{\frac {L}{L}}{\Bigr )}$ | $\Lambda _{d}={\frac {B^{2}}{2\ \rho \ (d^{2}/t)\ \mu _{0}\ \Omega }}{\Bigl{\frac {L}{L}}{\Bigr )}$ | | |
| Ordenando | $\Lambda _{d}={\frac {B^{2}}{2\ \rho \ \mu _{0}\ (L/t)\ (d^{2}/L)\ \Omega }}$ | $\Lambda _{d}={\frac {B^{2}}{2\ \rho \ \mu _{0}\ (L/t)\ (d^{2}/L)\ \Omega }}$ | | |
| Sustituyendo | $\Lambda _{d}={\frac {B^{2}}{2\ \rho \ \mu _{0}\ u\ \ell \ \Omega }}$ | $\Lambda _{d}={\frac {B^{2}}{2\ \rho \ \mu _{0}\ u\ \ell \ \Omega }}$ | $\Lambda _{d}={\frac {B^{2}}{2\ \rho \ \mu _{0}\ u\ \ell \ \Omega }}$ | $\Lambda _{d}={\frac {B^{2}}{2\ \rho \ \mu _{0}\ u\ \ell \ \Omega }}$ |

$\Lambda _{d}={\frac {B^{2}}{2\ \rho \ \mu _{0}\ u\ \ell \ \Omega }}$

Deducción :

| | 1 | 2 | 3 |
| - | - | - | - |
| Ecuaciones | $\Lambda _{d}={\frac {B^{2}}{2\ \rho \ \mu _{0}\ u\ \ell \ \Omega }}$ | $\Lambda _{i}={\frac {B^{2}}{2\ \rho \ \eta \ \mu _{0}\ \Omega }}$ | $\mathrm {R_{m}} ={\frac {u\ d}{\eta }}$ |
| Multiplicando ${\Bigl{\frac {\eta \ d}{\eta \ d}}{\Bigr )}$ | $\Lambda _{d}={\frac {B^{2}}{2\ \rho \ \mu _{0}\ u\ \ell \ \Omega }}{\Bigl{\frac {\eta \ d}{\eta \ d}}{\Bigr )}$ | | |
| Ordenando | $\Lambda _{d}={\frac {B^{2}}{2\ \rho \ (\eta )\ \mu _{0}\ \Omega }}{\Bigl{\frac {d}{[(u\ d)/\eta ]\ \ell }}{\Bigr )}$ | | |
| Sustituyendo | $\Lambda _{d}=\Lambda _{i}{\Bigl{\frac {d}{\mathrm {R_{m}} \ \ell }}{\Bigr )}$ | $\Lambda _{d}=\Lambda _{i}{\Bigl{\frac {d}{\mathrm {R_{m}} \ \ell }}{\Bigr )}$ | $\Lambda _{d}=\Lambda _{i}{\Bigl{\frac {d}{\mathrm {R_{m}} \ \ell }}{\Bigr )}$ |

$\Lambda _{d}=\Lambda _{i}{\Bigl{\frac {d}{\mathrm {R_{m}} \ \ell }}{\Bigr )}$

