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# 物理代写|热力学代写Thermodynamics代考|ENME485 Stability Versus Kirchhoff’s and Korteweg-Helmholtz’ Principles

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## 物理代写|热力学代写Thermodynamics代考|Stability Versus Kirchhoff’s and Korteweg-Helmholtz’ Principles

The analogy between Kirchhoff’s and Korteweg-Helmholtz case outlined in Sects. 5.3.1 and 5.3.7, respectively, allows us to limit ourselves to Kirchhoff’s case. If $\nabla T=0$ then $T=T_0$ and we may take $\mathbf{X}=\mathbf{E}+\mathbf{v} \wedge \mathbf{B}$ and $\mathbf{Y}=\frac{\mathbf{j}{e l}}{T_0}$ as the vector of thermodynamic forces and thermodynamic flows, respectively. Moreover, LNET holds for Joule heating, hence $0 \geq \frac{d_X P}{d t}=\frac{d_Y P}{d t}=\frac{1}{2} \frac{d P}{d t}$ and a stable steady state enjoys Lyapunov stability with $P$ as Lyapunov function. In the following, we are going to check under which condition the inequality is satisfied, i.e. under which condition the existence of a stable steady state agrees with $\nabla T=0$. To this purpose, we limit ourselves to the nonrelativistic limit where $\nabla \wedge \mathbf{B}=\mu_0 \mathbf{j}{e l}$, take the curl of both sides of this equation, invoke Ohm’s law, take into account that $\nabla \wedge \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$ and obtain the so-called ‘magnetic diffusion equation’:
$$\frac{\partial \mathbf{B}}{\partial t}-\nabla \wedge(\mathbf{v} \wedge \mathbf{B})=\frac{1}{\mu_0 \sigma_{\Omega}} \Delta \mathbf{B}$$
If $\sigma_{\Omega} \rightarrow \infty$ then the magnetic diffusion equation reduces to the condition for the conservation of magnetic flux $\frac{\partial \mathbf{B}}{\partial t}-\nabla \wedge(\mathbf{v} \wedge \mathbf{B})=0$, just like the vorticity balance $\frac{\partial}{\partial t}(\nabla \wedge \mathbf{v})-\nabla \wedge[\mathbf{v} \wedge(\nabla \wedge \mathbf{v})]=\nu \Delta(\nabla \wedge \mathbf{v})$ reduces in the $\nu \rightarrow 0$ limit to the condition for the conservation of the flux of vorticity (the circuitation) $\frac{\partial}{\partial t}(\nabla \wedge \mathbf{v})-\nabla \wedge[\mathbf{v} \wedge(\nabla \wedge \mathbf{v})]=0$. Both $\nu$ and $\frac{1}{\mu_0 \sigma_{\Omega}}$ have the dimensions of a diffusion coefficient, i.e. (length) ${ }^2$.(time) ${ }^{-1}$. Now, let $l$ be a typical length of the system. We define the dimensionless ‘Reynolds’ number’ $R e \equiv \frac{|\mathbf{v}| l}{\nu}$ and the dimensionless ‘magnetic Reynolds’ number’ $R e_M \equiv|\mathbf{v}| l \mu_0 \sigma_{\Omega}$. If $R e \gg 1(\ll 1)$ then the typical time scale $\frac{l^2}{\nu}$ of viscous dissipation is $\gg(\ll)$ the inertial time scale $\frac{l}{|\mathbf{v}|}$. If $R e_M \gg 1(\ll 1)$ then the typical time scale $l^2 \mu_0 \sigma_{\Omega}$ of Joule dissipation is $\gg(\ll)$ the inertial time scale $\frac{l}{|\mathbf{v}|}$. Let us check whether the sign of $\frac{d_Y P}{d t}$ is actually non-positive, as predicted by LNET. We obtain ${ }^{32}$ :
\begin{aligned} \frac{d_Y P}{d t} &=\frac{1}{T_0} \int(\mathbf{E}+\mathbf{v} \wedge \mathbf{B}) \cdot \frac{d \mathbf{j}{e l}}{d t} d \mathbf{x} \ &=\frac{1}{T_0} \int(\mathbf{E}+\mathbf{v} \wedge \mathbf{B}) \cdot \frac{\partial \mathbf{j}{e l}}{\partial t} d \mathbf{x}+O\left(\operatorname{Re}M\right) \ &=\frac{1}{\mu_0 T_0} \int(\mathbf{E}+\mathbf{v} \wedge \mathbf{B}) \cdot \nabla \wedge \frac{\partial \mathbf{B}}{\partial t} d \mathbf{x}+O\left(\operatorname{Re}_M\right) \ &=\frac{1}{\mu_0 T_0} \int \frac{\partial \mathbf{B}}{\partial t} \cdot \nabla \wedge(\mathbf{E}+\mathbf{v} \wedge \mathbf{B}) d \mathbf{x}+O\left(\operatorname{Re}_M\right) \ &=\frac{1}{\mu_0 T_0 \sigma{\Omega}} \int \frac{\partial \mathbf{B}}{\partial t} \cdot \nabla \wedge \mathbf{j}{e l} d \mathbf{x}+O\left(\operatorname{Re}_M\right) \ &=\frac{1}{\mu_0^2 T_0 \sigma{\Omega}} \int \frac{\partial \mathbf{B}}{\partial t} \cdot \nabla \wedge \nabla \wedge \mathbf{B} d \mathbf{x}+O\left(\operatorname{Re}M\right) \ &=-\frac{1}{\mu_0^2 T_0 \sigma{\Omega}} \int \frac{\partial \mathbf{B}}{\partial t} \cdot \Delta \mathbf{B} d \mathbf{x}+O\left(\operatorname{Re}_M\right) \ &=-\frac{1}{\mu_0 T_0} \int\left|\frac{\partial \mathbf{B}}{\partial t}\right|^2 d \mathbf{x}+O\left(\operatorname{Re}_M\right) \end{aligned}

## 物理代写|热力学代写Thermodynamics代考|Convection at Moderate Ra

Here we take advantage of the results of Sect. 5.3.1 and 5.3.7 and provide a qualitative discussion of a variational principle for convection cells, provided by Chandrasekhar in his rigorous, seminal discussion of the heat transport processes occurring in stars [39].

Convection occurs in a gravitating fluid under the effect of a temperature gradient: if the dimensionless Rayleigh number $R a$ (an increasing function of the absolute value of $|\nabla T|$ ) exceeds a threshold value $R a_{t h r}$ then buoyancy drives the spontaneous formation of convective motions, counteracted by dissipative effects like viscous heating and (in electrically conducting, magnetized fluids) Joule heating. A small mass element of a fluid in a constant gravitational field and with a nonuniform temperature gradient may undergo a rotational motion in convective cells because of buoyancy. The notion of a steady state is therefore meaningful in an averaged sense only, where the averaged is performed on both space and time. In all cases, in a steady state, the averaged amount of energy $\varepsilon_\nu$ transformed into heat per unit volume and time by viscous heating and Joule heating is equal to the averaged amount of energy $\varepsilon_g$ supplied per unit volume and time by the buoyancy to the fluid
$$\varepsilon_\nu=\varepsilon_g$$
Unless $|\nabla T|$ is very large, $\frac{T}{|\nabla T|} \gg$ the linear size of a small mass element. Consequently, the temperature jump inside a small mass element is negligible, at least if $|\nabla T|$ is not too large across the system, or, in other words, if $R a-R a_{t h r}$ is assumed to be not too large. Moreover, even if convection may locally affect the fluid density, we neglect the impact of this perturbation on the gravitational field, which may therefore be assumed to be constant at all times; it follows that-as shown in Sect. 4.2.7-gravity leaves entropy unaffected. Together, these assumptions allow us to apply both Korteweg-Helmholtz’ principle and Kirchhoff’s principle locally, hence: $\varepsilon_\nu=$ min. Together with the relationship above, this leads to:
$$\varepsilon_\nu=\min \quad \text { with the constraint } \varepsilon_\nu=\varepsilon_g$$

Now, convection is ruled by the interplay of gravity and $\nabla T$. Accordingly, if convection cells are present then $\varepsilon_g$ is an increasing function of $|\nabla T|$, i.e.: $\frac{d \varepsilon_g}{|\nabla T|}>0$. Since $R a \propto|\nabla T|$, the variational principle above reduces therefore to a variational principle which we refer to as ‘Chandrasekhar’s principle’ in the following:
$R a=\min \quad$ with the constraint $\varepsilon_\nu=\varepsilon_g$
Quoting Chandrasekhar [39] instability occurs at the minimum temperature gradient at which a balance can be maintained between the kinetic energy dissipated by viscosity and the internal energy released by the buoyancy force. Remarkably, we killed more birds with one stone, as Chandrasekhar’s principle applies equally well to fluids with viscous heating only, Joule heating only or with a mixture of both. This fact makes it uniquely useful when it comes to convection problems in geophysics and stellar physics. In all cases, it allows the selection of solutions of the equations of motion for the convective cells which are stable against perturbations and are therefore likely to be observed in Nature.

## 物理代写|热力学代写Thermodynamics代考|Stability Versus Kirchhoff’s and Korteweg-Helmholtz’ Principles

Kirchhoff 和 Korteweg-Helmholtz 安例之间的类比在 sects 中概述。5.3.1 和 $5.3 .7$ 分别允许我们将目己限制在 Kirchhoff 的案 例中。如果 $\nabla T=0$ 然后 $T=T_0$ 我们可能会采取 $\mathbf{X}=\mathbf{E}+\mathbf{v} \wedge \mathbf{B}$ 和 $\mathbf{Y}=\frac{\text { jel }}{T 0}$ 分别作为热力学力和热力学流的向量。此外，
LNET 适用于焦耳加热，因此 $0 \geq \frac{d X P}{d t}=\frac{d Y P}{d t}=\frac{1}{2} \frac{d P}{d t}$ 个个稳定的稳态具有 Lyapunov 稳定性 $P$ 作为李雅普诺夫函数。下面，

$$\frac{\partial \mathbf{B}}{\partial t}-\nabla \wedge(\mathbf{v} \wedge \mathbf{B})=\frac{1}{\mu_0 \sigma_{\Omega}} \Delta \mathbf{B}$$

$\frac{\partial}{\partial t}(\nabla \wedge \mathbf{v})-\nabla \wedge[\mathbf{v} \wedge(\nabla \wedge \mathbf{v})]=\nu \Delta(\nabla \wedge \mathbf{v})$ 椷少在 $\nu \rightarrow 0$ 限制涡量通量守恒的条件 (回路)
$\frac{\partial}{\partial t}(\nabla \wedge \mathbf{v})-\nabla \wedge[\mathbf{v} \wedge(\nabla \wedge \mathbf{v})]=0$. 两个都 $\nu$ 和 $\frac{1}{\mu_0 \sigma_{\Omega}}$ 具有扩散系数的维度，即（长度） ${ }^2$ 。（时间) ${ }^{-1}$. 现在，让 $l$ 是系统的

$$\frac{d Y P}{d t}=\frac{1}{T_0} \int(\mathbf{E}+\mathbf{v} \wedge \mathbf{B}) \cdot \frac{d \mathbf{j} e l}{d t} d \mathbf{x} \quad=\frac{1}{T_0} \int(\mathbf{E}+\mathbf{v} \wedge \mathbf{B}) \cdot \frac{\partial \mathbf{j} e l}{\partial t} d \mathbf{x}+O(\operatorname{Re} M)=\frac{1}{\mu_0 T_0} \int(\mathbf{E}+\mathbf{v} \wedge \mathbf{B}) \cdot \nabla \wedge \frac{\partial \mathbf{B}}{\partial t} d \mathbf{x}+O(\operatorname{Re} M) \quad \frac{1}{\mu_0 T_0}$$

## 物理代写|热力学代写Thermodynamics代考|Convection at Moderate Ra

$$\varepsilon_\nu=\varepsilon_g$$

$$\varepsilon_\nu=\min \quad \text { with the constraint } \varepsilon_\nu=\varepsilon_g$$

$R a=\min$ 有约束 $\varepsilon_\nu=\varepsilon_g$

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