Nonlinear Effects on the Precessional Instability in Magnetized Turbulence
By means of direct numerical simulations (DNS), we study the impact of an imposed uniform magnetic field on precessing magnetohydrodynamic homogeneous turbulence with a unit magnetic Prandtl number. The base flow which can trigger the precessional instability consists of the superposition of a solid...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-12-01
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| Series: | Atmosphere |
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| Online Access: | https://www.mdpi.com/2073-4433/11/1/14 |
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doaj-76116c5c3e6b40a696fe6d7fb061eec1 |
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| record_format |
Article |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Abdelaziz Salhi Amor Khlifi Claude Cambon |
| spellingShingle |
Abdelaziz Salhi Amor Khlifi Claude Cambon Nonlinear Effects on the Precessional Instability in Magnetized Turbulence Atmosphere precession instabilities magnetohydrodynamics (mhd) turbulence spectra of kinetic and magnetic energies |
| author_facet |
Abdelaziz Salhi Amor Khlifi Claude Cambon |
| author_sort |
Abdelaziz Salhi |
| title |
Nonlinear Effects on the Precessional Instability in Magnetized Turbulence |
| title_short |
Nonlinear Effects on the Precessional Instability in Magnetized Turbulence |
| title_full |
Nonlinear Effects on the Precessional Instability in Magnetized Turbulence |
| title_fullStr |
Nonlinear Effects on the Precessional Instability in Magnetized Turbulence |
| title_full_unstemmed |
Nonlinear Effects on the Precessional Instability in Magnetized Turbulence |
| title_sort |
nonlinear effects on the precessional instability in magnetized turbulence |
| publisher |
MDPI AG |
| series |
Atmosphere |
| issn |
2073-4433 |
| publishDate |
2019-12-01 |
| description |
By means of direct numerical simulations (DNS), we study the impact of an imposed uniform magnetic field on precessing magnetohydrodynamic homogeneous turbulence with a unit magnetic Prandtl number. The base flow which can trigger the precessional instability consists of the superposition of a solid-body rotation around the vertical (<inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> axis (with rate <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Ω</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and a plane shear (with rate <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>2</mn> <mi>ε</mi> <mo>Ω</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> viewed in a frame rotating (with rate <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mo>Ω</mo> <mi>p</mi> </msub> <mrow> <mo>=</mo> <mi>ε</mi> <mo>Ω</mo> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> about an axis normal to the plane of shear and to the solid-body rotation axis and under an imposed magnetic field that aligns with the solid-body rotation axis (<inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="bold-italic">B</mi> <mo>‖</mo> <mo mathvariant="bold">Ω</mo> <mo>)</mo> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> While rotation rate and Poincaré number are fixed, <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <mn>20</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.17</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> the <i><b>B</b></i> intensity was varied, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mspace width="4pt"></mspace> <mn>0.5</mn> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2.5</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> so that the Elsasser number is about <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Λ</mo> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mspace width="4pt"></mspace> <mn>2.5</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mn>62.5</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> respectively. At the final computational dimensionless time, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>t</mi> <mo>=</mo> <mn>2</mn> <mi>ε</mi> <mo>Ω</mo> <mi>t</mi> <mo>=</mo> <mn>67</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> the Rossby number Ro is about <inline-formula> <math display="inline"> <semantics> <mrow> <mn>0.1</mn> </mrow> </semantics> </math> </inline-formula> characterizing rapidly rotating flow. It is shown that the total (kinetic + magnetic) energy <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, production rate <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">P</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> due the basic flow and dissipation rate (<inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">D</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> occur in two main phases associated with different flow topologies: (i) an exponential growth and (ii) nonlinear saturation during which these global quantities remain almost time independent with <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">P</mi> <mo>∼</mo> <mi mathvariant="script">D</mi> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> The impact of a "strong" imposed magnetic field <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mn>2.5</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> on large scale structures at the saturation stage is reflected by the formation of structures that look like filaments and there is no dominance of horizontal motion over the vertical (along the solid-rotation axis) one. The comparison between the spectra of kinetic energy <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>E</mi> <mrow> <mo>(</mo> <mi>κ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>E</mi> <mrow> <mo>(</mo> <mi>κ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>,</mo> <msub> <mi>k</mi> <mo>‖</mo> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>E</mi> <mrow> <mi>κ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>,</mo> <msub> <mi>k</mi> <mo>‖</mo> </msub> <mo>=</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> at the saturation stage reveals that, at large horizontal scales, the major contribution to <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>E</mi> <mrow> <mo>(</mo> <mi>κ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> does not come only from the mode <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mo>‖</mo> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> but also from the <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mo>‖</mo> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> mode which is the most energetic. Only at very large horizontal scales at which <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>E</mi> <mrow> <mo>(</mo> <mi>κ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>)</mo> </mrow> <mo>∼</mo> <msubsup> <mi>E</mi> <mrow> <mn>2</mn> <mi>D</mi> </mrow> <mrow> <mo>(</mo> <mi>κ</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> the flow is almost two-dimensional. In the wavenumbers range <inline-formula> <math display="inline"> <semantics> <mrow> <mn>10</mn> <mo>≤</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>≤</mo> <mn>40</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> the spectra <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>E</mi> <mrow> <mo>(</mo> <mi>κ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>E</mi> <mrow> <mo>(</mo> <mi>κ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>,</mo> <msub> <mi>k</mi> <mo>‖</mo> </msub> <mo>=</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> respectively follow the scaling <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>k</mi> <mo>⟂</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>k</mi> <mo>⟂</mo> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msubsup> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> Unlike the velocity field the magnetic field remains strongly three-dimensional for all scales since <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>E</mi> <mrow> <mn>2</mn> <mi>D</mi> </mrow> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>)</mo> </mrow> <mo>≪</mo> <msup> <mi>E</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> At the saturation stage, the Alfvén ratio between kinetic and magnetic energies behaves like <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>k</mi> <mo>‖</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> </semantics> </math> </inline-formula> for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <msub> <mi>k</mi> <mo>‖</mo> </msub> <mo>/</mo> <mrow> <mo>(</mo> <mn>2</mn> <mi>ε</mi> <mo>Ω</mo> <mo>)</mo> </mrow> <mo><</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> |
| topic |
precession instabilities magnetohydrodynamics (mhd) turbulence spectra of kinetic and magnetic energies |
| url |
https://www.mdpi.com/2073-4433/11/1/14 |
| work_keys_str_mv |
AT abdelazizsalhi nonlineareffectsontheprecessionalinstabilityinmagnetizedturbulence AT amorkhlifi nonlineareffectsontheprecessionalinstabilityinmagnetizedturbulence AT claudecambon nonlineareffectsontheprecessionalinstabilityinmagnetizedturbulence |
| _version_ |
1724940197291884544 |
| spelling |
doaj-76116c5c3e6b40a696fe6d7fb061eec12020-11-25T02:04:56ZengMDPI AGAtmosphere2073-44332019-12-011111410.3390/atmos11010014atmos11010014Nonlinear Effects on the Precessional Instability in Magnetized TurbulenceAbdelaziz Salhi0Amor Khlifi1Claude Cambon2Département de Physique, Faculté des sciences de Tunis, Tunis 1060, TunisiaDépartement de Physique, Faculté des sciences de Tunis, Tunis 1060, TunisiaLaboratoire de Mécanique des Fluides et d’Acoustique, Université de Lyon, UMR 5509, Ecole Centrale de Lyon, CNRS, UCBL, CEDEX, INSA F-69134 Ecully, FranceBy means of direct numerical simulations (DNS), we study the impact of an imposed uniform magnetic field on precessing magnetohydrodynamic homogeneous turbulence with a unit magnetic Prandtl number. The base flow which can trigger the precessional instability consists of the superposition of a solid-body rotation around the vertical (<inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> axis (with rate <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Ω</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> and a plane shear (with rate <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>=</mo> <mn>2</mn> <mi>ε</mi> <mo>Ω</mo> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> viewed in a frame rotating (with rate <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mo>Ω</mo> <mi>p</mi> </msub> <mrow> <mo>=</mo> <mi>ε</mi> <mo>Ω</mo> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> about an axis normal to the plane of shear and to the solid-body rotation axis and under an imposed magnetic field that aligns with the solid-body rotation axis (<inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="bold-italic">B</mi> <mo>‖</mo> <mo mathvariant="bold">Ω</mo> <mo>)</mo> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> While rotation rate and Poincaré number are fixed, <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <mn>20</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.17</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> the <i><b>B</b></i> intensity was varied, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mspace width="4pt"></mspace> <mn>0.5</mn> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <mn>2.5</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> so that the Elsasser number is about <inline-formula> <math display="inline"> <semantics> <mrow> <mo>Λ</mo> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mspace width="4pt"></mspace> <mn>2.5</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mn>62.5</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> respectively. At the final computational dimensionless time, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>t</mi> <mo>=</mo> <mn>2</mn> <mi>ε</mi> <mo>Ω</mo> <mi>t</mi> <mo>=</mo> <mn>67</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> the Rossby number Ro is about <inline-formula> <math display="inline"> <semantics> <mrow> <mn>0.1</mn> </mrow> </semantics> </math> </inline-formula> characterizing rapidly rotating flow. It is shown that the total (kinetic + magnetic) energy <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, production rate <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi mathvariant="script">P</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> due the basic flow and dissipation rate (<inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">D</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> occur in two main phases associated with different flow topologies: (i) an exponential growth and (ii) nonlinear saturation during which these global quantities remain almost time independent with <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">P</mi> <mo>∼</mo> <mi mathvariant="script">D</mi> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> The impact of a "strong" imposed magnetic field <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mn>2.5</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> on large scale structures at the saturation stage is reflected by the formation of structures that look like filaments and there is no dominance of horizontal motion over the vertical (along the solid-rotation axis) one. The comparison between the spectra of kinetic energy <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>E</mi> <mrow> <mo>(</mo> <mi>κ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>E</mi> <mrow> <mo>(</mo> <mi>κ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>,</mo> <msub> <mi>k</mi> <mo>‖</mo> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>E</mi> <mrow> <mi>κ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>,</mo> <msub> <mi>k</mi> <mo>‖</mo> </msub> <mo>=</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> at the saturation stage reveals that, at large horizontal scales, the major contribution to <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>E</mi> <mrow> <mo>(</mo> <mi>κ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> does not come only from the mode <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mo>‖</mo> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> but also from the <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mo>‖</mo> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> mode which is the most energetic. Only at very large horizontal scales at which <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>E</mi> <mrow> <mo>(</mo> <mi>κ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>)</mo> </mrow> <mo>∼</mo> <msubsup> <mi>E</mi> <mrow> <mn>2</mn> <mi>D</mi> </mrow> <mrow> <mo>(</mo> <mi>κ</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> the flow is almost two-dimensional. In the wavenumbers range <inline-formula> <math display="inline"> <semantics> <mrow> <mn>10</mn> <mo>≤</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>≤</mo> <mn>40</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> the spectra <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>E</mi> <mrow> <mo>(</mo> <mi>κ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>E</mi> <mrow> <mo>(</mo> <mi>κ</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>,</mo> <msub> <mi>k</mi> <mo>‖</mo> </msub> <mo>=</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> respectively follow the scaling <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>k</mi> <mo>⟂</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>k</mi> <mo>⟂</mo> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msubsup> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> Unlike the velocity field the magnetic field remains strongly three-dimensional for all scales since <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>E</mi> <mrow> <mn>2</mn> <mi>D</mi> </mrow> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>)</mo> </mrow> <mo>≪</mo> <msup> <mi>E</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mo>⟂</mo> </msub> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> At the saturation stage, the Alfvén ratio between kinetic and magnetic energies behaves like <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>k</mi> <mo>‖</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> </semantics> </math> </inline-formula> for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>B</mi> <msub> <mi>k</mi> <mo>‖</mo> </msub> <mo>/</mo> <mrow> <mo>(</mo> <mn>2</mn> <mi>ε</mi> <mo>Ω</mo> <mo>)</mo> </mrow> <mo><</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics> </math> </inline-formula>https://www.mdpi.com/2073-4433/11/1/14precessioninstabilitiesmagnetohydrodynamics (mhd) turbulencespectra of kinetic and magnetic energies |