File Name: angular momentum problems and solutions .zip
Canuto 1 ,2. Received: 17 March Accepted: 10 October We present a formalism that provides the Reynolds stresses needed to solve the angular momentum equation.
The traditional Reynolds stress model assumes that the only contribution comes from shear a down-gradient flux , but this leads to an extraction of angular momentum from the interior that is far too small compared to what is required to explain the helio seismological data. An illustrative solution of the new Reynolds stress equations shows that the presence of vorticity in a stably stratified regime, such as the one in the radiative zone, contributes a new term to the angular momentum equation that has an up-gradient flux like the one provided by the IGW model internal gravity waves.
It would be instructive to solve the new angular momentum equation together with the formalism developed in Paper III to study not only the solar angular momentum distribution vs.
These results would allow a more quantitative assessment of the overall model. Helio-seismological data have presented us with an interesting new feature: solar rotation changes from differential in the convective zone CZ to uniform in the radiative, stably stratified region below the CZ. The problem has been studied for many years by many authors, and in what follows we present a brief summary of the present situation:.
The reason identified bythe authors of those studies is the limited range of physicalparameters compared to the true solar values allowed bynumerical simulations, e. Such models yield large rotation gradients in the stellar interiors that are not consistent with helio data;. For small Ri corresponding to a strong shear, as expected in the CZ-RZ transition zone, the time scale can be of the same order as the one provided by the IGW mechanism.
In the next sections, we first show why the angular momentum equation employed in all previous studies is an approximation of the complete one. We then derive and show how to include vorticity and stratification , which contribute with opposite signs in the CZ and in the RZ zones where they produce an up-gradient flux in the angular momentum equation.
Finally, we show how the new Reynolds stress model includes unstable stratification, stable stratification, differential rotation, double-diffusion, arbitrary Peclet number accounting for radiative losses , and meridional currents. The component of Eq. Use of Eqs. The first is the need to justify why only shear enters, and second is how one determines the momentum diffusivity. The first item is almost never discussed and the determination of K m is handled with heuristic models discussed in Sect.
This leads to 1a , which is not a diffusion equation, in spite of being generally referred to as such. In the next sections, we show that shear alone giving rise to 1a is not a justifiable approximation, and then we proceed to extend 1d to include vorticity, buoyancy, radiative losses, double diffusion, and meridional currents. Consider the first relation in Eq.
For a 3D flow, in general, the measured flow distribution can only be predicted by choosing different viscosities for each stress component Markatos Thus, in going from 1d to 2e , we have gone from 2i where S , V , B stand for shear, vorticity and buoyancy.
Other considerations are also in order. Since diffusion involves small scales, it is clear that shear, which is governed by large scales, cannot represent diffusion. Since small scales have large vorticity, it is only natural that the latter be present in 2i.
Second, to properly describe the qualitative difference in the rotation curves in the convective and radiative regimes, it is only natural to have the buoyancy flux, which is positive in the first and negative in the second regime.
Third, what about the Peclet number representing radiative losses? If we recall Eq. This is because both the eddy velocity and the length scale are large in the CZ but small in the stably stratified radiative regime. Thus, Eq. As shown above, they enter directly into the angular momentum Eqs. If we succeed in constructing 2k , we would have included shear, vorticity, different regimes of both unstable convective zone, CZ and stable stratification radiative regime, RZ , radiative losses, and meridional currents.
While there is no guarantee that the resulting rotational curve will explain the helio data, we would have at least made sure that we have included the key processes that characterize the two regimes of interest, CZ and RZ. It is fair to ask whether the RSM has been previously employed in a stellar context. None of the models with 3c were able to reproduce the measured surface data of 3b.
In particular, the first two combinations in 3c gave the wrong sign in both hemispheres. However, as already discussed, thus far all the solutions of the angular momentum equation, which yielded results in disagreement with helio data, were based on the same assumption, the first of 3c , which failed to reproduce the surface values of 3b.
Clearly, such a computation must be done in conjunction with a solar structure code to provide the mean variables. One can actually perform two computations, with and without double-diffusion, and compare the results, a process that would be quite instructive. As discussed in Sect. However, this is seldom done because of the added complexity.
See for example Eqs. In addition to Eq. The solutions represented by nested algebraic relations, were suited to a numerical treatment but not to physical considerations.
Here, we present a solution of the same equations with no meridional currents and no double diffusion, a simplification that allows highlighting a new feature of the model, the existence of a counter-gradient angular momentum flux within the hydrodynamic instability framework. We begin by introducing the following dimensionless variables: 9a The dimensionless form of Eq.
Several comments are in order:. Vorticity and buoyancy appear together in 10b , which issomething of a surprise since such a combination was not obviousin the starting Eqs. For comments, see the text. The first term also has a positive sign, and thus the two terms on the rhs in 13 both have a positive sign, which means that they are both of the counter-gradient type.
The first term in 13 is of the counter-gradient type, while the second term is up-gradient. It is instructive to compare the momentum diffusivity K m obtained in this work with the one used in the literature. Several comments about 15 are in order. The first is about the absence of a factor representing the amount of energy or power that creates the mixing. In fact, turbulence is a process that does not generate or destroy energy, rather, it distributes whatever energy is put into the system among a wide variety of scales.
Without such an energy, there would be no turbulent motion or, in the presence of turbulence, turning such source of energy off would lead to a decaying turbulent mixing. Thus, the presence of such a factor would ensure that different stars give rise to different states of mixing, as ought to be the case. The factor 16a is clearly different for different stars. On the other hand, in the present formalism, the momentum diffusivity can be written in a variety of forms beginning with Eq.
As already discussed in Sect. By comparison, if we use the standard Eq. Finally, let us take the last term in 13 corresponding to an up-gradient model UG. Given the well known challenge of trying to reproduce the helio data on the solar rotation curve, we have examined the ingredients of the angular momentum equation. It contains two main terms, the Reynolds stresses and the meridional currents which, in a large Re regime such as that characterizing a stellar interior and in a steady state, must balance each other.
It is worth noticing that this balance has not yet been exhibited by the numerical simulations published thus far see Figs. Since the equations for the meridional currents also depend on the Reynolds stresses, the latter constitute a key ingredient and we have therefore concentrated on how they have been modeled thus far and what the missing terms are that must be included.
The final new formula for the Reynolds stress is quite simple, Eq. This is a welcome feature if one considers the large amount of information that the new model contains: stable stratification, unstable stratification, double-diffusion, differential rotation, shear, radiative losses arbitrary Peclet number and meridional currents. As an illustrative example we have worked out the case of no double diffusion and no meridional currents so as to highlight a key feature of the model.
The standard RSM model, based on shear alone, is of the down-gradient type, and it fails to reproduce the helio data that point to a rigid body rotation below the solar convective zone. Here, we have introduced an alternative that we hasten to stress is not ad hoc but is part and parcel of the RSM model. The dimensionless functions S m in Eq. Data correspond to usage on the plateform after The current usage metrics is available hours after online publication and is updated daily on week days.
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Measurements and their descriptions Top Abstract 1. The solar Why shear alone? Previous results Reynolds stresses, Meridional currents 7.
Example: no Conclusions References List of figures. Brummell, N. Fluid Mech. Kupka, I. Chan Cambridge Univ. Press , IAU Symp. Current usage metrics About article metrics Return to article. Initial download of the metrics may take a while.
Why does Earth keep on spinning? What started it spinning to begin with? And how does an ice skater manage to spin faster and faster simply by pulling her arms in? Why does she not have to exert a torque to spin faster? Questions like these have answers based in angular momentum, the rotational analog to linear momentum. By now the pattern is clear—every rotational phenomenon has a direct translational analog. As we would expect, an object that has a large moment of inertia I , such as Earth, has a very large angular momentum.
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In an isolated system, the moment of inertia of a rotating object is halved. What happens to the angular velocity of the object? In an isolated system, there is no net torque. If there is no net torque on the system, then the total angular momentum of the system remains the same. The angular momentum of a rotating object is equal to the moment of inertia of the object multiplied by the object's angular velocity. This is because , which is the multiplicative identity.
the angular momentum is perpendicular to the plane formed by the position and momentum vectors. For this problem that means either into the paper, denoted.
So far, we have looked at the angular momentum of systems consisting of point particles and rigid bodies. We have also analyzed the torques involved, using the expression that relates the external net torque to the change in angular momentum, Equation In this case, Equation The angular momentum of a system of particles around a point in a fixed inertial reference frame is conserved if there is no net external torque around that point:. Any of the individual angular momenta can change as long as their sum remains constant.
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Canuto 1 ,2. Received: 17 March Accepted: 10 October We present a formalism that provides the Reynolds stresses needed to solve the angular momentum equation. The traditional Reynolds stress model assumes that the only contribution comes from shear a down-gradient flux , but this leads to an extraction of angular momentum from the interior that is far too small compared to what is required to explain the helio seismological data. An illustrative solution of the new Reynolds stress equations shows that the presence of vorticity in a stably stratified regime, such as the one in the radiative zone, contributes a new term to the angular momentum equation that has an up-gradient flux like the one provided by the IGW model internal gravity waves. It would be instructive to solve the new angular momentum equation together with the formalism developed in Paper III to study not only the solar angular momentum distribution vs. These results would allow a more quantitative assessment of the overall model.
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