Fig 4 - uploaded by Axel Brandenburg
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— Normalized time lag ∆t/τ versus ωτ for different values of ω 0 τ. Note the development of a peak near ωτ = 1 as ω 0 τ is increased. replace Equation (35) by: 1 + ω 2 0 τ 2 + 2τ
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In the mean-field theory of magnetic fields, turbulent transport, i.e. the turbulent electromotive force, is described by a combination of the alpha effect and turbulent magnetic diffusion, which are usually assumed to be proportional respectively to the mean field and its spatial derivatives. For a passive scalar there is just turbulent diffusion,...
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Citations
... Writing Γ X ,Φ as the product of the modulus and phase |Γ X ,Φ | exp( ), one can show that the heat flux response, at a monochromatic frequency , is related to the temperature gradient as (Hubbard and Brandenburg 2009), ...
In recent theory trying to explain the origin of baroclinic low-frequency atmospheric variability, the concept of eddy memory has been proposed. In this theory, the effect of synoptic-scale heat fluxes on the planetary-scale mean flow depends on the history of the mean meridional temperature gradient. Mathematically, this involves the convolution of a memory kernel with the mean meridional temperature gradient over past times. However, the precise shape of the memory kernel and its connection to baroclinic wave dynamics remains to be explained. In this study we use linear and proxy response theory to determine the shape of the memory kernel of a truncated two-layer quasigeostrophic atmospheric model. We find a memory kernel that relates the eddy heat flux to the zonal mean meridional temperature gradient on time scales greater than 2 days. Although the shape of the memory kernel is complex, we show that it may be well approximated as an exponential, particularly when reproducing baroclinic low-frequency intraseasonal modes of variability. By computing the terms in the Lorenz energy cycle, we find that the shape of the memory kernel can be linked to the finite time that growing baroclinic instabilities require to adapt their growth properties to the local zonal mean atmospheric flow stability. Regarding the explanation for observed baroclinic annular modes in the Southern Hemisphere, our results suggest that it is physical for these modes to be derived directly from the thermodynamic equation by considering an exponentially decaying memory kernel, provided accurate estimates of the necessary parameters are incorporated.
Significance Statement
The goal of this study was to derive the memory of the zonal mean temperature field contained in eddy heat fluxes. To do this we used recent developments in a theory stemming from statistical mechanics, called proxy response theory. This theory facilitated direct numerical computations of the parameterization that links eddy heat fluxes to the zonal mean temperature field. Notably, this parameterization incorporates a crucial memory component, which we demonstrated to be essential in explaining the periodicity of low-frequency modes of variability, specifically the baroclinic annular mode (BAM). Understanding the role of memory as a driver of this variability holds great significance, as the BAM constitutes a dominant pattern of large annular variability within the Southern Hemisphere circulation. Enhanced comprehension of this driver, which is memory, can lead to improved understanding and predictive capabilities concerning observed annular weather patterns.
... If the transport coefficients themselves nonlinearly depend on the inhomogeneous boundaries of the system, the deviation from the equilibrium state can be enhanced. The non-equilibrium properties of turbulence are sometimes represented by a memory effect on the transport coefficient (Hubbard & Brandenburg 2009). The memory effect based on the response function or time integral kernel gives further information on the turbulent transport beyond the simple argument based on the moments of correlation functions. ...
Turbulence is typically not in equilibrium, i.e. mean quantities such as the mean energy and helicity are typically time-dependent. The effect of non-stationarity on the turbulent hydromagnetic dynamo process is studied here with the use of the Two-Scale Direct-Interaction Approximation (TSDIA), which allows to explicitly relate the mean turbulent Reynolds and Maxwell stresses and the mean electromotive force (EMF) to the spectral characteristics of turbulence, such as e.g. the mean energy, as well as kinetic and cross-helicity. It is demonstrated, that the non-equilibrium effects can enhance the dynamo process when the magnetohydrodynamic (MHD) turbulence is both helical and cross-helical. This effect is based on the turbulent infinitesimal-impulse cross-response functions, which do not affect turbulent flows in equilibrium. The evolution and sources of the cross-helicity in MHD turbulence is also discussed.
... The authors of [6] showed that the memory effect strongly affects the dynamo action. The [7] used the formalism of response functions and showed that the effect of the integral kernels can be significant for anisotropic flows. Therefore, when modeling a dynamo, it is desirable to take into account this memory (heredity). ...
We study some fractal properties of the hereditary αω-dynamo model in the two-mode approximation. The phase variables of the model describe the temporal dynamics of the toroidal and poloidal components of the magnetic field. The hereditary operator of the quenching the α-effect by field helicity in numerical simulation is determined using the Riemann–Liouville fractional differentiation operator. The model also includes a stochastic term. The structure of this term corresponds to the effect of coherent structures from small-scale magnetic field and velocity modes. A difference scheme and a program code for numerical simulation have been developed and verified. A series of computational experiments with the model has been carried out. The Hausdorff dimension of the polarity scale in the model and the distribution of polarity intervals are calculated. It is shown that the Hausdorff dimension of the polarity scale is less than 1, i.e., this scale is a fractal. The numerical value of the dimension for some values of the control parameters is 0.87, which is consistent with the dimension of the real geomagnetic polarity scale. The distribution histogram of polarity intervals in the model has a pronounced power-law tail, which also agrees with the properties of real polarity scales.
... Both the spatial nonlocality (see Brandenburg & Sokoloff 2002) of the dynamo closure relation, the noninstantaneous aspects (i.e., so-called "memory effects"; e.g., Hubbard & Brandenburg 2009), as well as their combined effect (see, e.g., Rheinhardt & Brandenburg 2012) have been demonstrated to influence the characteristics of the dynamo cycle. Another comprehensive example of how finite-time effects can influence dynamogenerated fields has been presented by Chamandy et al. (2013aChamandy et al. ( , 2013b in the context of galactic magnetic fields. ...
... Typically, the turbulent closure coefficients are thought to connect z t , ( ) to the mean magnetic field, B z t , ( ), and its curl B z t , jzl z l ( ) e ¶ , in a local and instantaneous fashion. 4 However, in contrast to this instantaneous characterization of the closure relation, the power-law nature of the turbulent cascade suggests that the spacetime domain of dependence of z t , ( ) is indeed finite-implying so-called "memory effects" (Hubbard & Brandenburg 2009), that is, a delayed (i.e., out-of-phase) response to an applied mean field. 5 Under the assumption of statistically stationary turbulence, a simple noninstantaneous closure relation (also see can be formulated as a convolution integral in time of the form ...
... Moreover, in its Fourier-space representation, the above relation can be expressed as a simple multiplication (see Hubbard & Brandenburg 2009, Appendix A) with the Fourier transform, ij a , of the kernel. That is, (dropping the explicit z-dependence) we write ...
Accretion disk turbulence along with its effect on large-scale magnetic fields plays an important role in understanding disk evolution in general, and the launching of astrophysical jets in particular. Motivated by enabling a comprehensive subgrid description for global long-term simulations of accretions disks, we aim to further characterize the transport coefficients emerging in local simulations of magnetorotational disk turbulence. For the current investigation, we leverage a time-dependent version of the test-field method, which is sensitive to the turbulent electromotive force (EMF) generated as a response to a set of pulsating background fields. We obtain Fourier spectra of the transport coefficients as a function of oscillation frequency. These are well approximated by a simple response function, describing a finite-time buildup of the EMF as a result of a time-variable mean magnetic field. For intermediate timescales (i.e., slightly above the orbital frequency), we observe a significant phase lag of the EMF compared to the causing field. Augmented with our previous result on a nonlocal closure relation in space, and incorporated into a suitable mean-field description that we briefly sketch out here, the new framework will allow us to drop the restrictive assumption of scale separation.
... Both the spatial non-locality (see Brandenburg & Sokoloff 2002) of the dynamo closure relation, the non-instantaneous aspects (i.e., so-called "memory effects", e.g., Hubbard & Brandenburg 2009), as well as their combined effect (see, e.g., Rheinhardt & Brandenburg 2012) have been demonstrated to influence the characteristics of the dynamo cycle. Another comprehensive example of how finite-time effects can influence dynamo-generated fields has been presented by Chamandy et al. (2013a,b) in the context of galactic magnetic fields. ...
... Typically, the turbulent closure coefficients are thought to connect E(z, t) to the mean magnetic field, B(z, t), and its curl ε jzl ∂ z B l (z, t), in a local and instantaneous fashion. 2 How-ever, in contrast to this instantaneous characterization of the closure relation, the power-law nature of the turbulent cascade suggests that the space-time domain of dependence of E(z, t) is indeed finite -implying so-called "memory effects" (Hubbard & Brandenburg 2009), that is, a delayed (i.e., outof-phase) response to an applied mean field. 3 Under the assumption of statistically stationary turbulence, a simple noninstantaneous closure relation (also see can be formulated as a convolution integral in time of the form ...
... In the local box geometry, the integral kernelsα ij (z, t ) and η ij (z, t ) are functions of the vertical coordinate, z, only. Moreover, in its Fourier-space representation, the above relation can be expressed as a simple multiplication (see Hubbard & Brandenburg 2009, appendix A) with the Fourier transform,α ij , of the kernel. That is (dropping the explicit z-dependence), we writẽ ...
Accretion disc turbulence along with its effect on large-scale magnetic fields plays an important role in understanding disc evolution in general, and the launching of astrophysical jets in particular. Motivated by enabling a comprehensive sub-grid description for global long-term simulations of accretions discs, we aim to further characterize the transport coefficients emerging in local simulations of magnetorotational disc turbulence. For the current investigation, we leverage a time-dependent version of the test-field method, which is sensitive to the turbulent electromotive force (EMF) generated as a response to a set of pulsating background fields. We obtain Fourier spectra of the transport coefficients as a function of oscillation frequency. These are well approximated by a simple response function, describing a finite-time build-up of the EMF as a result of a time-variable mean magnetic field. For intermediate timescales (i.e., slightly above the orbital frequency), we observe a significant phase lag of the EMF compared to the causing field. Augmented with our previous result on a non-local closure relation in space, and incorporated into a suitable mean-field description that we briefly sketch out here, the new framework will allow to drop the restrictive assumption of scale separation.
... One of the simplest approach is called the minimal approximation, where the third-order momentum represented as a forcing for the time evolution of second-order moments is approximated as a damping term with the time-scale (Brandenburg et al., 2004;Brandenburg and Subramanian, 2005). This is equivalent to applying an integral kernel instead of a constant diffusivity with a finite memory (Hubbard and Brandenburg, 2009). ...
The baroclinic annular mode (BAM) is a leading‐order mode of the eddy kinetic energy in the Southern Hemisphere exhibiting oscillatory behaviour at intraseasonal time‐scales. The oscillation mechanism has been linked to transient eddy–mean flow interactions which remain poorly understood. Here we demonstrate that the finite memory effect in eddy‐heat flux dependence on the large‐scale flow can explain the origin of the BAM's oscillatory behaviour. We represent the eddy memory effect by a delayed integral kernel that leads to a generalized Langevin equation for the planetary‐scale heat equation. Using a mathematical framework for the interactions between planetary‐ and synoptic‐scale motions, we derive a reduced dynamical model of the BAM – a stochastically forced oscillator with a period proportional to the geometric mean between the eddy memory time‐scale and the diffusive eddy equilibration time‐scale. Our model provides a formal justification for the previously proposed phenomenological model of the BAM and could be used to explicitly diagnose the memory kernel and improve our understanding of transient eddy–mean flow interactions in the atmosphere.
... Now, we can take the planetary-scale spatial average over the equations (21,22), and combine the two by replacing w 1 by the horizontal vorticity convergence, which leads to ...
... One of the simplest approach is called the minimal τ approximation, where the third-order momentum represented as a forcing for the time-evolution of second-order moments is approximated as a damping term with the timescale τ [5,6]. This is equivalent to apply an integral kernel instead of a constant diffusivity with a finite memory [21]. ...
The baroclinic annular mode (BAM) is a leading-order mode of the eddy-kinetic energy in the Southern Hemisphere exhibiting. oscillatory behavior at intra-seasonal time scales. The oscillation mechanism has been linked to transient eddy-mean flow interactions that remain poorly understood. Here we demonstrate that the finite memory effect in eddy-heat flux dependence on the large-scale flow can explain the origin of the BAM's oscillatory behavior. We represent the eddy memory effect by a delayed integral kernel that leads to a generalized Langevin equation for the planetary-scale heat equation. Using a mathematical framework for the interactions between planetary and synoptic-scale motions, we derive a reduced dynamical model of the BAM - a stochastically-forced oscillator with a period proportional to the geometric mean between the eddy-memory time scale and the diffusive eddy equilibration timescale. Our model provides a formal justification for the previously proposed phenomenological model of the BAM and could be used to explicitly diagnose the memory kernel and improve our understanding of transient eddy-mean flow interactions in the atmosphere.
... In the simplest cases, this feedback is considered to be instantaneous in time and local in space. However, the correct description of turbulent transfer may include both spatial nonlocality [13] and memory effect [14][15][16]. The consideration of memory in low-mode dynamo models can affect essentially the dynamic regimes. ...
Cosmic magnetic fields possess complex time dynamics. They are characterized by abrupt polarity changes (reversals), fluctuations of fixed polarity, bursts and attenuations. These dynamic conditions can replace each other, including both regular and chaotic components. Memory in dynamo systems manifests itself in a feedback mechanism when a strong magnetic field begins to change the properties of turbulent flows. A hereditary oscillator can be the simplest model of such complex oscillatory systems with memory. The article suggests the construction of such oscillator by means of two-mode approximation of magnetic field components in the αω-dynamo model. The hereditary member describes the suppression of a field turbulent generator by magnetic helicity and determines the shape of oscillator potential. The article describes the implicit difference scheme for numerical research of oscillator. It also describes the results of numerical simulation for two cases—instantaneous feedback and delay in feedback. The results of simulation are interpreted in terms of oscillator theory. It is shown that the observed dynamic regimes in the model go well with the change of potential shape.
... Both methods gave a good approximation of the EMF by the mean magnetic field, but with slightly different values for the turbulent diffusivity. We take now test fields with variations in space (Brandenburg, Rädler & Schrinner 2008) and time (Hubbard & Brandenburg 2009) for the analysis of the turbulent EMF. Unlike in earlier studies, we can investigate the action of turbulence on various length scales in the kinematic regime of the dynamo, and study the importance of so-called "memory effects". ...
... Conversely to this local interpretation of the closure, the multiscale character of turbulence suggests that the domain of dependence of E(z, t) is finite. This entails both non-local dependencies in space, as well as so-called "memory effects" (e.g., Hubbard & Brandenburg 2009). ...
... For this closure relation, Hubbard & Brandenburg (2009) have demonstrated that, to a reasonable degree of approximation, one can model the frequency-dependence as an "oscillating decay" of the form ∝ Θ(t) e −t/τc cos(ω 0 t), where Θ(t) denotes the Heaviside step function. When translated to Fourier space, this implies spectral dependencies of the form ...
The interstellar medium of the Milky Way and nearby disk galaxies harbours large-scale coherent magnetic fields of Microgauss strength, that can be explained via the action of a mean-field dynamo. As in our previous work, we aim to quantify dynamo effects that are self-consistently emerging in realistic direct magnetohydrodynamic simulations, but we generalise our approach to the case of a non-local (non-instantaneous) closure relation, described by a convolution integral in space (time). To this end, we leverage our comprehensive simulation framework for the supernova-regulated turbulent multi-phase interstellar medium. By introducing spatially (temporally) modulated mean fields, we extend the previously used test-field method to the spectral realm -- providing the Fourier representation of the convolution kernels. The resulting spectra of the dynamo mean-field coefficients that we obtain broadly match expectations and allow to rigorously constrain the degree of scale separation in the Galactic dynamo. A surprising result is found for the diamagnetic pumping term, which increases in amplitude when going to smaller scales. Our results amount to the most comprehensive description of dynamo mean-field effects in the Galactic context to date. Surveying the relevant parameter space and quenching behaviour, this will ultimately enable the development of assumption-free sub-grid prescriptions for otherwise unresolved global galaxy simulations.
... Theη xx component determines the evolution of A x and correspondingly B y , which is decaying even in the marginally excited case; see figure 2. The time dependence of B leads a memory effect Mean-field generation in optimal dynamos 15 FIGURE 9. Dependence ofη xx (red) andη yy (blue) on k for the Willis flow in the marginally exited case with η = 0.403 and for ω = 0. The dashed line denotes the fit −0.233 + 0.11k 2 and will be discussed in § 7.2. in the evolution of A x , i.e. to a frequency-dependent time delay in the electromotive force (Hubbard & Brandenburg 2009). By contrast,η yy is real for time-independent mean magnetic fields, but it is negative for ω → 0 and k → 0; see figure 9. ...
In recent years, several optimal dynamos have been discovered. They minimize the magnetic energy dissipation or, equivalently, maximize the growth rate at a fixed magnetic Reynolds number. In the optimal dynamo of Willis ( Phys. Rev. Lett. , vol. 109, 2012, 251101), we find mean-field dynamo action for planar averages. One component of the magnetic field grows exponentially while the other decays in an oscillatory fashion near onset. This behaviour is different from that of an $\unicode[STIX]{x1D6FC}^{2}$ dynamo, where the two non-vanishing components of the planar averages are coupled and have the same growth rate. For the Willis dynamo, we find that the mean field is excited by a negative turbulent magnetic diffusivity, which has a non-uniform spatial profile near onset. The temporal oscillations in the decaying component are caused by the corresponding component of the diffusivity tensor being complex when the mean field is decaying and, in this way, time dependent. The growing mean field can be modelled by a negative magnetic diffusivity combined with a positive magnetic hyperdiffusivity. In two other classes of optimal dynamos of Chen et al. ( J. Fluid Mech. , vol. 783, 2015, pp. 23–45), we find, to some extent, similar mean-field dynamo actions. When the magnetic boundary conditions are mixed, the two components of the planar averaged field grow at different rates when the dynamo is 15 % supercritical. When the mean magnetic field satisfies homogeneous boundary conditions (where the magnetic field is tangential to the boundary), mean-field dynamo action is found for one-dimensional averages, but not for planar averages. Despite having different spatial profiles, both dynamos show negative turbulent magnetic diffusivities. Our finding suggests that negative turbulent magnetic diffusivities may support a broader class of dynamos than previously thought, including these three optimal dynamos.
