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On the spatial segregation of helicity by inertial waves in dynamo simulations and planetary cores

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Davidson, PA 

Abstract

jats:pThe distribution of kinetic helicity in a dipolar planetary dynamo is central to the success of that dynamo. Motivated by the helicity distributions observed in numerical simulations of the Earth’s dynamo, we consider the relationship between the kinetic helicity, jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline1" />jats:tex-math𝜵h=u⋅𝜵×u</jats:tex-math></jats:alternatives></jats:inline-formula>, and the buoyancy field that acts as a source of helicity, where jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline2" />jats:tex-mathu</jats:tex-math></jats:alternatives></jats:inline-formula> is velocity. We show that, in the absence of a magnetic field, helicity evolves in accordance with the equation jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline3" />jats:tex-math𝜵h/t=−𝜵⋅F+Sh</jats:tex-math></jats:alternatives></jats:inline-formula>, where the flux, jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline4" />jats:tex-mathF</jats:tex-math></jats:alternatives></jats:inline-formula>, represents the transport of helicity by inertial waves, and the helicity source, jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline5" />jats:tex-mathSh</jats:tex-math></jats:alternatives></jats:inline-formula>, involves the product of the buoyancy and the velocity fields. In the numerical simulations it is observed that the helicity outside the tangent cylinder is predominantly negative in the north and positive in the south, a feature which the authors had previously attributed to the transport of helicity by waves (Davidson & Ranjan, jats:italicGeophys. J. Intl</jats:italic>, vol. 202, 2015, pp. 1646–1662). It is also observed that there is a strong spatial correlation between the distribution of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline6" />jats:tex-mathh</jats:tex-math></jats:alternatives></jats:inline-formula> and of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline7" />jats:tex-mathSh</jats:tex-math></jats:alternatives></jats:inline-formula>, with jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline8" />jats:tex-mathSh</jats:tex-math></jats:alternatives></jats:inline-formula> also predominantly negative in the north and positive in the south. This correlation tentatively suggests that it is the jats:italicin situ</jats:italic> generation of helicity by buoyancy that establishes the distribution of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline9" />jats:tex-mathh</jats:tex-math></jats:alternatives></jats:inline-formula> outside the tangent cylinder, rather than the dispersal of helicity by waves, as had been previously argued by the authors. However, although jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline10" />jats:tex-mathh</jats:tex-math></jats:alternatives></jats:inline-formula> and jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline11" />jats:tex-mathSh</jats:tex-math></jats:alternatives></jats:inline-formula> are strongly correlated, there is no such correlation between jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline12" />jats:tex-mathh/t</jats:tex-math></jats:alternatives></jats:inline-formula> and jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline13" />jats:tex-mathSh</jats:tex-math></jats:alternatives></jats:inline-formula>, as might be expected if the distribution of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline14" />jats:tex-mathh</jats:tex-math></jats:alternatives></jats:inline-formula> were established by an jats:italicin situ</jats:italic> generation mechanism. We explain these various observations by showing that inertial waves interact with the buoyancy field in such a way as to induce a source jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline15" />jats:tex-mathSh</jats:tex-math></jats:alternatives></jats:inline-formula> which has the same sign as the helicity in the local wave flux, and that the sign of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline16" />jats:tex-mathh</jats:tex-math></jats:alternatives></jats:inline-formula> is simply determined by the direction of that flux. We conclude that the observed distributions of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline17" />jats:tex-mathh</jats:tex-math></jats:alternatives></jats:inline-formula> and jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018004974_inline18" />jats:tex-mathSh</jats:tex-math></jats:alternatives></jats:inline-formula> outside the tangent cylinder are consistent with the transport of helicity by waves.</jats:p>

Description

Keywords

geodynamo, geophysical and geological flows, waves in rotating fluids

Journal Title

Journal of Fluid Mechanics

Conference Name

Journal ISSN

0022-1120
1469-7645

Volume Title

851

Publisher

Cambridge University Press (CUP)