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.^[1]​

Simbología

Simbología

Símbolo	Nombre	Unidad
${\displaystyle \Lambda _{i}}$	Número de Elsasser impuesto
${\displaystyle \Lambda _{d}}$	Número de Elsasser dinámico
${\displaystyle \mathrm {Ek} }$	Número de Ekman
${\displaystyle \mathrm {P_{m}} }$	Número de Prandtl magnético
${\displaystyle \mathrm {R_{m}} }$	Número de Reynolds magnético
	Dimensiones
${\displaystyle \ell }$		m
${\displaystyle L}$	Longitud	m
${\displaystyle d}$	Dimensión de sección transversal	m
${\displaystyle t}$	Tiempo	s
	Magnético
${\displaystyle B}$	Densidad de flujo magnético	T
${\displaystyle H}$	Fuerza de campo magnético	A / m
${\displaystyle \rho _{m}}$	Densidad de energía magnética	J / m3
${\displaystyle q}$	Carga	C
${\displaystyle \eta }$	Difusividad magnética	m2 / s
${\displaystyle \mu _{0}}$	Permeabilidad en el vacío	H / m
${\displaystyle \theta }$	Ángulo entre velocidad (${\displaystyle u}$) y campo magnético (${\displaystyle B}$)
	Materia
${\displaystyle m}$	Masa	kg
${\displaystyle {\dot {m}}}$	Flujo másico	kg / s
${\displaystyle u}$	Velocidad	m / s
${\displaystyle \rho }$	Densidad	kg / m3
${\displaystyle \nu }$	Viscosidad cinemática	m2 / s
	Planeta
${\displaystyle \phi }$	Latitud
${\displaystyle \Omega }$	Velocidad de rotación del cuerpo	m / s

Descripción

Se define como:

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

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

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

Deducción

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

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

Deducción

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

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

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

Deducción

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

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

Deducción

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

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

Referencias

Bibliografía

• Gubbins, David, and Emilio Herrero-Bervera, eds. Encyclopedia of geomagnetism and paleomagnetism. Springer Science & Business Media, 2007.