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Astrophysical Dynamo Theory David Hughes . Department of Applied Mathematics, University of Leeds 1/47 Astrophysical Dynamos • What are they? • Why do we need them? • Why are they difficult to study? • What are the outstanding problems? 2/47 The Geomagnetic Field The Earth has had a magnetic field for at least 3 billion years. The field retains one polarity for long periods of time, interspersed with fairly rapid reversals. The temporal distribution of the reversals appears to be random. 3/47 Why do we need dynamo theory? Consideration of the Earth’s magnetic field provides one of the clearest answers to this question. 4/47 Why do we need dynamo theory? Consideration of the Earth’s magnetic field provides one of the clearest answers to this question. At the most basic level, there are two ways of explaining the continued existence of a magnetic field in an astrophysical body. 4/47 Why do we need dynamo theory? Consideration of the Earth’s magnetic field provides one of the clearest answers to this question. At the most basic level, there are two ways of explaining the continued existence of a magnetic field in an astrophysical body. 1. The field has been there since the body was formed. 4/47 Why do we need dynamo theory? Consideration of the Earth’s magnetic field provides one of the clearest answers to this question. At the most basic level, there are two ways of explaining the continued existence of a magnetic field in an astrophysical body. 1. The field has been there since the body was formed. or ....... 2. It hasn’t! 4/47 Why do we need dynamo theory? Consideration of the Earth’s magnetic field provides one of the clearest answers to this question. At the most basic level, there are two ways of explaining the continued existence of a magnetic field in an astrophysical body. 1. The field has been there since the body was formed. or ....... 2. It hasn’t! In the absence of motion, magnetic field satisfies the diffusion equation ∂B = η∇2 B, ∂t with Ohmic timescale of decay T = L2 /η (here L is characteristic length, η is magnetic diffusivity). 4/47 Why do we need dynamo theory? Consideration of the Earth’s magnetic field provides one of the clearest answers to this question. At the most basic level, there are two ways of explaining the continued existence of a magnetic field in an astrophysical body. 1. The field has been there since the body was formed. or ....... 2. It hasn’t! In the absence of motion, magnetic field satisfies the diffusion equation ∂B = η∇2 B, ∂t with Ohmic timescale of decay T = L2 /η (here L is characteristic length, η is magnetic diffusivity). For the Earth, T ∼ 104 years, whereas the field has existed for 3 × 109 years. Thus the field cannot be a fossil field. Clearly, inductive motions are crucial for the maintenance of the field. 4/47 The Solar Magnetic Field 1919: Joseph Larmor: ‘Brief Communication’ to the British Association for the Advancement of Science, How Could a Rotating Body Such as the Sun Become a Magnet? This might be thought of as the starting point for astrophysical dynamo theory. The Sun’s magnetic field can be observed over a wide range of scales and is responsible for most solar dynamic phenomena. 5/47 Large-Scale Solar Field Magnetogram of the entire Sun. Black and white regions denote regions of strong positive and negative polarity — in visible light these regions are sunspots. Suggestive of a strong toroidal (east-west) field bursting through the solar surface. Note that the field is antisymmetric about the equator. 6/47 Solar Cycle 7/47 Solar Cycle Temporal variation of sunspot number since 1600. 7/47 Solar Cycle The field reverses sign approximately every 11 years. 8/47 Large-Scale Solar Field: Summary of Key Observations • The field is predominantly dipolar. • It has a strong toroidal (azimuthal) component, antisymmetric about the equator. • The azimuthal field is confined to lower latitudes and propagates towards the equator, with an approximate period of 11 years. • The field reverses every 11 years, so a full magnetic cycle is 22 years. • The amplitude of the field is modulated in time in a complicated manner (e.g. the Maunder minimum). A full theory of the solar magnetic field must explain these observations. 9/47 Large-Scale Solar Field: Summary of Key Observations • The field is predominantly dipolar. • It has a strong toroidal (azimuthal) component, antisymmetric about the equator. • The azimuthal field is confined to lower latitudes and propagates towards the equator, with an approximate period of 11 years. • The field reverses every 11 years, so a full magnetic cycle is 22 years. • The amplitude of the field is modulated in time in a complicated manner (e.g. the Maunder minimum). A full theory of the solar magnetic field must explain these observations. It is of interest to note that the Ohmic decay time for the Sun is long, unlike that of the Earth. Indeed L2 /η ∼ 109 years, comparable to the lifetime of the Sun. Thus we cannot rule out a fossil field explanation for the Sun simply on these grounds. However, it is very difficult to explain the short term variations in terms of a fossil field. 9/47 Small-Scale Solar Magnetic Fields Solar granulation. Bright spots are the sites of intense magnetic field. Small-scale field has no preferred orientation and is not correlated with the solar cycle. Suggests that this field is self-maintained, rather than being just a by-product of the large-scale dynamo. 10/47 Stellar Magnetic Fields Magnetic field activity detected on other solar-like stars via Ca II emission (e.g. Baliunas et al 1995). Cyclic activity found on fairly slow rotators (such as the Sun). More vigorous, but less egular behavour found on rapid rotators. 11/47 Dynamo Experiments Figure: Karlsruhe experiment side view Figure: Karlsruhe top view Figure: VKS experiment Figure: Madison plasma experiment 12/47 The Governing Equations The dynamics of the magnetic field in stellar interiors is well described by the equations of single fluid magnetohydrodynamics (MHD): 13/47 The Governing Equations The dynamics of the magnetic field in stellar interiors is well described by the equations of single fluid magnetohydrodynamics (MHD): Induction equation: ∂B = ∇ × (u × B) + η∇2 B. ∂t Momentum equation: ∂u + u · ∇u = −∇p + j × B + ρg + F other + F viscous . ρ ∂t Mass conservation: ∂ρ + ∇ · (ρu) = 0. ∂t Energy equation: D pρ−γ = loss terms. Dt Equation of state: p = RρT . 13/47 Dynamo Terminology Theoreticians find it helpful to classify dynamos into various categories — although Nature has no need to obey this classification. For example, dynamos can be classified as: (i) Kinematic or dynamic (ii) Slow or fast, (iii) Small scale or large scale. 14/47 Kinematic Dynamos Kinematic dynamo theory addresses the following (simply stated) question: Can we find a velocity field u(x, t) such that the magnetic field — governed solely by the induction equation — grows? 15/47 Kinematic Dynamos Kinematic dynamo theory addresses the following (simply stated) question: Can we find a velocity field u(x, t) such that the magnetic field — governed solely by the induction equation — grows? So here the problem is reduced to just one equation, linear in the magnetic field B: ∂B = ∇ × (u × B) + η∇2 B. ∂t R Magnetic energy M(t) = B 2 dV . A velocity field u(x, t) acts as a (kinematic) dynamo if M(t) 9 0 as t → ∞. 15/47 Kinematic Dynamos Kinematic dynamo theory addresses the following (simply stated) question: Can we find a velocity field u(x, t) such that the magnetic field — governed solely by the induction equation — grows? So here the problem is reduced to just one equation, linear in the magnetic field B: ∂B = ∇ × (u × B) + η∇2 B. ∂t R Magnetic energy M(t) = B 2 dV . A velocity field u(x, t) acts as a (kinematic) dynamo if M(t) 9 0 as t → ∞. Induction – leads to growth of energy through extension of field lines. Dissipation – leads to decay of energy into heat through Ohmic loss. Hydrodynamic dynamo works if induction by the fluid motions overpowers the dissipative losses. 15/47 Kinematic Dynamos Kinematic dynamo theory addresses the following (simply stated) question: Can we find a velocity field u(x, t) such that the magnetic field — governed solely by the induction equation — grows? So here the problem is reduced to just one equation, linear in the magnetic field B: ∂B = ∇ × (u × B) + η∇2 B. ∂t R Magnetic energy M(t) = B 2 dV . A velocity field u(x, t) acts as a (kinematic) dynamo if M(t) 9 0 as t → ∞. Induction – leads to growth of energy through extension of field lines. Dissipation – leads to decay of energy into heat through Ohmic loss. Hydrodynamic dynamo works if induction by the fluid motions overpowers the dissipative losses. It is not straightforward to find rigorous examples of flows that act as dynamos. Most of the early work demonstrated flows and fields that could not act as dynamos. 15/47 Anti-Dynamo Results, or Necessary Conditions for Dynamo Action These can be divided into three categories: (i) Constraints on the symmetries of magnetic fields that cannot be generated by dynamo action. (ii) Constraints on the symmetries of velocity fields that cannot act as dynamos. (iii) Lower bounds on the magnetic Reynolds number (or similar quantities) that must be exceeded for dynamo action. 16/47 Anti-Dynamo Results, or Necessary Conditions for Dynamo Action These can be divided into three categories: (i) Constraints on the symmetries of magnetic fields that cannot be generated by dynamo action. (ii) Constraints on the symmetries of velocity fields that cannot act as dynamos. (iii) Lower bounds on the magnetic Reynolds number (or similar quantities) that must be exceeded for dynamo action. The most famous anti-dynamo theorem (which is in category (i)) is Cowling’s (1934) theorem: An axisymmetric magnetic field cannot be maintained by dynamo action. 16/47 Anti-Dynamo Theorems (ii) 1957: Zeldovich theorem. Magnetic fields cannot be maintained by two-dimensional planar motions. (Also true for motions on a spherical surface, but not for motions on a cylindrical surface.) 17/47 Anti-Dynamo Theorems (ii) 1957: Zeldovich theorem. Magnetic fields cannot be maintained by two-dimensional planar motions. (Also true for motions on a spherical surface, but not for motions on a cylindrical surface.) Proof Consider flows u(x, t) with u · ẑ = 0; for simplicity suppose the flow is incompressible. Write B = B · ẑ . Then DB = η∇2 B Dt and so B decays everywhere to zero (multiply by B and integrate over the domain). So we need consider B only in the xy -plane. Write B = ∇ × Aẑ . Then the induction equation becomes DA = η∇2 A. Dt Thus it follows also that A must decay. Thus no magnetic field can be maintained by a purely two dimensional planar flow. Note that this result holds even if B is z-dependent. 17/47 Anti-Dynamo Theorems (ii) 1957: Zeldovich theorem. Magnetic fields cannot be maintained by two-dimensional planar motions. (Also true for motions on a spherical surface, but not for motions on a cylindrical surface.) Proof Consider flows u(x, t) with u · ẑ = 0; for simplicity suppose the flow is incompressible. Write B = B · ẑ . Then DB = η∇2 B Dt and so B decays everywhere to zero (multiply by B and integrate over the domain). So we need consider B only in the xy -plane. Write B = ∇ × Aẑ . Then the induction equation becomes DA = η∇2 A. Dt Thus it follows also that A must decay. Thus no magnetic field can be maintained by a purely two dimensional planar flow. Note that this result holds even if B is z-dependent. Moral of the story: Too much symmetry is a bad thing. 17/47 Anti-Dynamo Theorems (iii) Backus (1958): A necessary condition for dynamo action driven by a steady flow in a sphere of radius R is that em R 2 /η > π 2 , where em is the maximum rate of strain. The left hand-side could be regarded as a magnetic Reynolds number (here defined unconventionally using the rate of strain rather than the velocity). 18/47 Anti-Dynamo Theorems (iii) Backus (1958): A necessary condition for dynamo action driven by a steady flow in a sphere of radius R is that em R 2 /η > π 2 , where em is the maximum rate of strain. The left hand-side could be regarded as a magnetic Reynolds number (here defined unconventionally using the rate of strain rather than the velocity). Childress (1969) proved a further necessary condition , namely Um R/η > π 2 , where here Um is the maximum velocity in the flow. So using the conventional definition of magnetic Reynolds number, Rm = UL/η, we could write this as max(Rm) > π 2 . 18/47 First Working Dynamo: Herzenberg (1958) Spheres radius a, separation R, rotation axes inclined at angle φ. Toroidal field from sphere (1) diffuses to sphere (2) where it is, at least partially, poloidal. It is then wound up to give a toroidal field at (2). Field diffuses to (1) — where it has a poloidal component since rotation axes not parallel — and can then be wound up by rotation of (1) to give toroidal component. 19/47 First Working Dynamo: Herzenberg (1958) Spheres radius a, separation R, rotation axes inclined at angle φ. Toroidal field from sphere (1) diffuses to sphere (2) where it is, at least partially, poloidal. It is then wound up to give a toroidal field at (2). Field diffuses to (1) — where it has a poloidal component since rotation axes not parallel — and can then be wound up by rotation of (1) to give toroidal component. Complicated analysis shows dynamo action possible if Rm > 15(R/a)3 cos φ sin2 φ if R a. 19/47 First Working Dynamo: Herzenberg (1958) Spheres radius a, separation R, rotation axes inclined at angle φ. Toroidal field from sphere (1) diffuses to sphere (2) where it is, at least partially, poloidal. It is then wound up to give a toroidal field at (2). Field diffuses to (1) — where it has a poloidal component since rotation axes not parallel — and can then be wound up by rotation of (1) to give toroidal component. Complicated analysis shows dynamo action possible if Rm > 15(R/a)3 cos φ sin2 φ if R a. First conclusive demonstration of dynamo action — though not relevant to isolated astrophysical bodies. 19/47 Dynamo Classification From an astrophysical point of view, we are often interested in the generation of large-scale fields (such as that on the Sun). Also, astrophysically the magnetic Reynolds number Rm is huge, so we are interested in dynamo action at high Rm. 20/47 Dynamo Classification From an astrophysical point of view, we are often interested in the generation of large-scale fields (such as that on the Sun). Also, astrophysically the magnetic Reynolds number Rm is huge, so we are interested in dynamo action at high Rm. These two problems have essentially been studied separately. (i) The large-scale dynamo problem has typically been studied using mean field electrodynamics. (ii) The high Rm limit has led to what is known as the fast dynamo problem. 20/47 Mean Field Magnetohydrodynamics The generation of large-scale fields is typically studied using Mean Field MHD. This is, at heart, a kinematic (linear) theory, which is often then modified to include nonlinear effects. The linear theory is self-consistent (though is not without problems); the nonlinear theory not necessarily so. 21/47 Kinematic mean field electrodynamics Magnetic field is governed by the induction equation ∂B = ∇ × (u × B) + η∇2 B, ∂t where B is magnetic field, u is fluid velocity, η is magnetic diffusivity. 22/47 Kinematic mean field electrodynamics Magnetic field is governed by the induction equation ∂B = ∇ × (u × B) + η∇2 B, ∂t where B is magnetic field, u is fluid velocity, η is magnetic diffusivity. Standard formulation splits velocity and magnetic field into mean (large-scale) and fluctuating (small-scale) parts: U = U 0 + u, B = B 0 + b. 22/47 Kinematic mean field electrodynamics Magnetic field is governed by the induction equation ∂B = ∇ × (u × B) + η∇2 B, ∂t where B is magnetic field, u is fluid velocity, η is magnetic diffusivity. Standard formulation splits velocity and magnetic field into mean (large-scale) and fluctuating (small-scale) parts: U = U 0 + u, B = B 0 + b. Averaging the induction equation leads to the following equations for the mean and fluctuating magnetic fields: ∂B 0 = ∇ × (U 0 × B 0 ) + ∇ × E + η∇2 B 0 , ∂t ∂b = ∇ × (U 0 × b) + ∇ × (u × B 0 ) + ∇ × G + η∇2 b, ∂t where mean emf E = hu × bi and G = (u × b) − hu × bi. 22/47 Relation between mean field and mean emf Equation for the fluctuating field can be written as L(b) = ∇ × (u × B 0 ) . Linear relation between b and B 0 implies linear relation between E and B 0 . 23/47 Relation between mean field and mean emf Equation for the fluctuating field can be written as L(b) = ∇ × (u × B 0 ) . Linear relation between b and B 0 implies linear relation between E and B 0 . Mean emf is then written as the expansion: Ei = αij B0j + βijk ∂B0j + ··· , ∂xk where convergence is anticipated as a result of the large spatial scale of B 0 . 23/47 Relation between mean field and mean emf Equation for the fluctuating field can be written as L(b) = ∇ × (u × B 0 ) . Linear relation between b and B 0 implies linear relation between E and B 0 . Mean emf is then written as the expansion: Ei = αij B0j + βijk ∂B0j + ··· , ∂xk where convergence is anticipated as a result of the large spatial scale of B 0 . In its simplest, kinematic, formulation, the α coefficients can be calculated by measuring E after imposing a uniform kinematic magnetic field on the small-scale flow. The αij depend on the properties of u and on η and provide information on the growth of large-scale magnetic fields. 23/47 The Mean Field Induction Equation Consider the simplest case of isotropic turbulence. Then αij = αδij and βijk = βijk . Substituting for the mean emf gives the mean induction equation ∂B 0 = ∇ × (U 0 × B 0 ) + ∇ × (αB 0 ) + (η + β)∇2 B. ∂t So β (in this simplest case) can be regarded as a turbulent diffusivity. The more interesting term is that involving α, which is of a completely different form to any term in the unaveraged induction equation. 24/47 How does the new term work? Decompose the magnetic field into poloidal (B P ) and toroidal (B T ) components. Toroidal field can be produced from poloidal field by differential rotation pulling out a poloidal field (so-called Omega-effect — the easy bit). 25/47 How does the new term work? Decompose the magnetic field into poloidal (B P ) and toroidal (B T ) components. Toroidal field can be produced from poloidal field by differential rotation pulling out a poloidal field (so-called Omega-effect — the easy bit). Mathematically the dynamo loop can be closed (B T → B P ) through the α term (the ‘α-effect’). Physically one may think of this picture: A poloidal field is raised and twisted, givng rise to a tooidal current. Averaging over many such events leads to a net J T and hence a net B P (Parker’s (1955) ‘cyclonic events’). 25/47 Parker Dynamo Waves Simplest possible dynamo model consists of a plane layer in Cartesian geometry, with a velocity shear (to generate toroidal from poloidal field) and an α-effect (to close the loop and generate poloidal from toroidal). Suppose x points south, y points east and z vertically upwards. Seek an ‘axisymmetric’ field B = B(x, z, t)ŷ + ∇ × (A(x, z, t)ŷ ) . Then the mean field induction equation becomes ∂A = αB + η∇2 A, ∂t ∂B ∂A = V0 + η∇2 B, ∂t ∂x where V is the mean azimuthal velocity (the differential rotation). For simplicity, assume V 0 and α are constants. Then we can seek plane wave solutions A = Â exp(pt + ikx), B = B̂ exp(pt + ikx). 26/47 Parker Dynamo Waves Leads to the dispersion relation √ p = −ηk 2 1 ± (1 + i) D , where the dynamo number D = αV 0 /2ηk 3 . There are growing solutions (i.e. dynamo action) if |D| > 1. Importantly, these modes propagate as dynamo waves. Propagation is towards the equator if αV 0 < 0 and towards the poles if αV 0 > 0. Such a dynamo is known as an αω- dynamo (driven by a combination of the α effect and the differential rotation ω. It provides the simplest explanation of the solar butterfly diagram. 27/47 More Complicated Mean Field Modelling Going beyond the simple Parker waves, by choosing suitable forms of α, β and the differential rotation (possibly with ad hoc nonlinearities), it is possible to model magnetic fields in a range of astrophysical bodies. (Dikpati et al 2006) 28/47 More Complicated Mean Field Modelling Going beyond the simple Parker waves, by choosing suitable forms of α, β and the differential rotation (possibly with ad hoc nonlinearities), it is possible to model magnetic fields in a range of astrophysical bodies. (Dikpati et al 2006) But note that this is not the same as these results being rigorously derived, and is a weakness of the mean field theory. One can obtain similar output with very different models, and very different output with quite similar models. Mean field models have been tuned to the last few solar cycles and predictions made for the current solar cycle 24 — but these have proved to be very wide of the mark. 28/47 Two other serious difficulties 1. Results obtained for small Rm or small S do not carry over to the true astrophysical regime (Rm 1, S = O(1)). (a) makes sense for short correlation times or lots of diffusion. (b) (or something more complicated) holds otherwise. 29/47 An example: mean emf from rotating convection (Hughes & Cattaneo JFM, 2008). Top plot shows time sequence of already spatially averaged emf. Bottom plot shows the cumulative average. Slow convergence to an extremely small value of α. α here scales as urms /Rm, rather than with urms as one would expect from a turbulent (fast) process. 30/47 Small Scale Dynamo Growth 2. L(b) = ∇ × (u × B 0 ) . The standard formulation assumes there are no exponentially growing solutions to the equation L(b) = 0, i.e. no small-scale dynamo action. However there may be — and at high Rm there almost certainly will be. 31/47 Small Scale Dynamo Growth 2. L(b) = ∇ × (u × B 0 ) . The standard formulation assumes there are no exponentially growing solutions to the equation L(b) = 0, i.e. no small-scale dynamo action. However there may be — and at high Rm there almost certainly will be. This leads us on nicely to ..... 31/47 Fast Dynamo Theory Astrophysically, the magnetic Reynolds number Rm is invariably extremely large. Mathematically this has led to the idea of looking at kinematic dynamos in the limit as Rm → ∞. Suppose that a flow u(x, t) acts as a dynamo for some finite Rm. Then if dynamo action persists as Rm → ∞, the dynamo is said to be fast. Otherwise it is said to be slow. All of the fast dynamos considered have been small-scale dynamos. 32/47 Anti-dynamo theorem for fast dynamos The main theorem, due to Vishik and Klapper & Young states that: The dynamo growth rate is bounded above by the topological entropy of the flow. 33/47 Anti-dynamo theorem for fast dynamos The main theorem, due to Vishik and Klapper & Young states that: The dynamo growth rate is bounded above by the topological entropy of the flow. The topological entropy can here be identified as the exponential growth of material lines — and importantly is zero for a non-chaotic flow. Thus, as a corollary, we have the important anti-dynamo theorem: Fast dynamo action cannot arise as a result of an integrable flow. i.e. a necessary condition for fast dynamo action is that the flow be chaotic (i.e. have exponentially separating trajectories). 33/47 Chaotic Flows Flows depending on just two variables, x and y say, are computationally helpful since the z-dependence of the magnetic field is then just exp(ikz). For a given k the problem involves only two, rather than three, spatial dimensions. 34/47 Chaotic Flows Flows depending on just two variables, x and y say, are computationally helpful since the z-dependence of the magnetic field is then just exp(ikz). For a given k the problem involves only two, rather than three, spatial dimensions. Steady 2-d flows are integrable and hence cannot be fast. But flows u(x, y , t) are good candidates for study if they are chaotic. 34/47 Chaotic Flows Flows depending on just two variables, x and y say, are computationally helpful since the z-dependence of the magnetic field is then just exp(ikz). For a given k the problem involves only two, rather than three, spatial dimensions. Steady 2-d flows are integrable and hence cannot be fast. But flows u(x, y , t) are good candidates for study if they are chaotic. The aim is to produce some kind of evidence for the growth rate as Rm → ∞. Not easy numerically of course; 3d steady and unsteady flows are even more demanding computationally. 34/47 The Galloway Proctor flow GP-flow (Nature, 1992): u = ∇ × (ψ(x, y , t)ẑ ) + ψ(x, y , t)ẑ ψ(x, y , t) = A (cos(x + cos t) + sin(y + sin t)) 35/47 The Galloway Proctor flow GP-flow (Nature, 1992): u = ∇ × (ψ(x, y , t)ẑ ) + ψ(x, y , t)ẑ ψ(x, y , t) = A (cos(x + cos t) + sin(y + sin t)) (GP flow movie) 35/47 Chaos in the GP flow Finite-time Lyapunov exponents showing the stretching of an infinitesimal element initially located at the marked location. An infinitesimal element ξ satisfies dξ = ξ · u. dt The Lyapunov exponent is an average measure of the exponential stretching, along a fluid trajectory. 36/47 Growth rates Growth rates versus k at fixed Rm. 37/47 Growth rates Growth rates versus k at fixed Rm. Growth rates versus Rm at fixed k. (a) is the GP flow. 37/47 Growth rates Growth rates versus k at fixed Rm. Growth rates versus Rm at fixed k. (a) is the GP flow. Good numerical evidence of self-similar behaviour with constant growth rate as Rm → ∞. But not a proof! 37/47 Numerical Modelling Most of what I have discussed so far either addresses the linear (kinematic) problem or incorporates nonlinearities into the mean field coefficients. The full dynamo problem of course requires a self-consistent solution to all the governing equations, which requires a computational approach. 38/47 Numerical Modelling Most of what I have discussed so far either addresses the linear (kinematic) problem or incorporates nonlinearities into the mean field coefficients. The full dynamo problem of course requires a self-consistent solution to all the governing equations, which requires a computational approach. It should though be pointed out that astrophysical parameters are either very small or very large, so it is impossible to perform totally realistic astrophysical simulations. At the base of the solar convection zone, the various dimensionless parameters are: Ra = g (∇ − ∇ad )L4 /(νκHp ) Re = UL/ν Rm = UL/η Pr = ν/κ Pm = ν/η β = pgas /pmag M = U/cs Ro = U/2ΩL Rayleigh no. Reynolds no. magnetic Reynolds no. Prandtl no. magnetic Prandtl no. plasma beta Mach no. Rossby no. 1020 1013 1010 10−7 10−3 105 10−4 10−1 (A similar problem arises in geodynamo modelling: the Earth is a rapid rotator, with Ekman number E = ν/2ΩL2 = O(10−15 .) 38/47 Numerical Modelling Some of the difficulties of the extreme parameter values can be addressed through modifications to the governing equations. 39/47 Numerical Modelling Some of the difficulties of the extreme parameter values can be addressed through modifications to the governing equations. For example, tracking sound waves numerically is very computationally expensive, since the waves are fast and hence the timestep is small. If these are not dynamically important for the physical process being considered then they can be removed, via an asymptotic reduction of the equations. The Boussinesq approximation neglects sound waves and treats the fluid as incompressible. The anelastic approximation, which is more general but also more complicated, also neglects sound waves but retains the effects of density stratification. Most numerical dynamo models solve the anelastic equation in a rotating spherical shell. 39/47 Numerical Modelling Some of the difficulties of the extreme parameter values can be addressed through modifications to the governing equations. For example, tracking sound waves numerically is very computationally expensive, since the waves are fast and hence the timestep is small. If these are not dynamically important for the physical process being considered then they can be removed, via an asymptotic reduction of the equations. The Boussinesq approximation neglects sound waves and treats the fluid as incompressible. The anelastic approximation, which is more general but also more complicated, also neglects sound waves but retains the effects of density stratification. Most numerical dynamo models solve the anelastic equation in a rotating spherical shell. That said, dealing with the issues of high Re and Rm and low Pm remain in place. It is worth noting that in the largest current simulations one can get to Re ∼ Rm ∼ 104 — still a long way short of the true values! 39/47 Numerical Modelling Some of the difficulties of the extreme parameter values can be addressed through modifications to the governing equations. 40/47 Numerical Modelling Some of the difficulties of the extreme parameter values can be addressed through modifications to the governing equations. For example, tracking sound waves numerically is very computationally expensive, since the waves are fast and hence the timestep is small. If these are not dynamically important for the physical process being considered then they can be removed, via an asymptotic reduction of the equations. The Boussinesq approximation neglects sound waves and treats the fluid as incompressible. The anelastic approximation, which is more general but also more complicated, also neglects sound waves but retains the effects of density stratification. Most numerical dynamo models solve the anelastic equation in a rotating spherical shell. 40/47 Numerical Modelling Some of the difficulties of the extreme parameter values can be addressed through modifications to the governing equations. For example, tracking sound waves numerically is very computationally expensive, since the waves are fast and hence the timestep is small. If these are not dynamically important for the physical process being considered then they can be removed, via an asymptotic reduction of the equations. The Boussinesq approximation neglects sound waves and treats the fluid as incompressible. The anelastic approximation, which is more general but also more complicated, also neglects sound waves but retains the effects of density stratification. Most numerical dynamo models solve the anelastic equation in a rotating spherical shell. That said, dealing with the issues of high Re and Rm and low Pm remain in place. It is worth noting that in the largest current simulations one can get to Re ∼ Rm ∼ 104 — still a long way short of the true values! 40/47 Computational Dynamo Models Simulations of anelastic convection, increasing the rotation rate (from Brown et al 2008 Ap.J.). 41/47 Computational Dynamo Models Toroidal and radial magnetic field for moderate rotation rate. Note that the field is essentially small scale (from Brun et al 2004 Ap.J.). 42/47 Computational Dynamo Models At very high rotation rates the field takes the from of ‘magnetic wreaths’ (from Brown et al 2010 Ap.J.). 43/47 Current Views on the Solar Dynamo Internal solar rotation, deduced from helioseismology. The thin white region between the convective and radiative zones is the tachocline, a region of strong radial shear. 44/47 Current Views on the Solar Dynamo Internal solar rotation, deduced from helioseismology. The thin white region between the convective and radiative zones is the tachocline, a region of strong radial shear. There are currently three models for the solar dynamo; all have problems. 44/47 Current Views on the Solar Dynamo Internal solar rotation, deduced from helioseismology. The thin white region between the convective and radiative zones is the tachocline, a region of strong radial shear. There are currently three models for the solar dynamo; all have problems. (i) Dynamo action distributed throughout the convection zone. 44/47 Current Views on the Solar Dynamo Internal solar rotation, deduced from helioseismology. The thin white region between the convective and radiative zones is the tachocline, a region of strong radial shear. There are currently three models for the solar dynamo; all have problems. (i) Dynamo action distributed throughout the convection zone. (ii) An interface dynamo in which dynamo action takes place in, and just above, the tachocline. 44/47 Current Views on the Solar Dynamo Internal solar rotation, deduced from helioseismology. The thin white region between the convective and radiative zones is the tachocline, a region of strong radial shear. There are currently three models for the solar dynamo; all have problems. (i) Dynamo action distributed throughout the convection zone. (ii) An interface dynamo in which dynamo action takes place in, and just above, the tachocline. (iii) Flux transport dynamo, in which dynamo action takes place in two widely separated regions: the tachocline and the surface. 44/47 Distributed Dynamo • Here the poloidal field is generated throughout the convection zone by the action of cyclonic turbulence. • Toroidal field is generated by the latitudinal distribution of differential rotation. • No role is envisaged for the tachocline Pros Scenario is theoretically possible wherever convection and rotation take place together. Cons Small-scale dynamo action may dominate. Indeed, computations at high Rm show that it is hard to generate a large-scale field. 45/47 Interface Dynamo • The dynamo is thought to work at the interface of the convection zone and the tachocline. • The mean toroidal (sunspot field) is created by the radial differential rotation and stored in the tachocline. • The mean poloidal field (coronal field) is created by turbulence (some sort of α-effect) in the lower reaches of the convection zone. Pros The radial shear provides a natural mechanism for generating a strong toroidal field. The stable stratification enables the field to be stored and stretched to a large value. As the mean magnetic field is stored away from the convection zone, the α-effect is not suppressed by a strong field. Cons Relies on transport of flux (of just the right strength) to and from tachocline ? how is this achieved? 46/47 Flux Transport Dynamo • Here the poloidal field is generated at the surface of the Sun via the decay of active regions with a systematic tilt (Babcock-Leighton scenario) and transported towards the poles by the observed meridional flow. • The flux is then transported by a conveyor belt meridional flow to the tachocline where it is sheared into the sunspot toroidal field. • No role whatsoever is envisaged for the turbulent convection in the bulk of the convection zone. Pros Does not rely on turbulent α-effect. Sunspot field is intimately linked to polar field immediately before. Cons Requires strong meridional flow at base of CZ of exactly the right form. Ignores all poloidal flux returned to tachocline via the convection Effect will probably be swamped by ‘α-effects’ closer to the tachocline. Relies on existence of sunspots for dynamo to work (cf. Maunder Minimum). 47/47