Dynamics II

Lecture: April 8, 2024 (Monday), 14:00 Prof. Dr. Gerrit Lohmann

 
 
 

Preparation

Chapter The equations of fluid motion (Marchal and Plumb, 2008)

Video Introduction to Atmospheric Dynamics (47 min)

Basics in Chapter 1 “The Equations of Atmospheric Dynamics” Holton and Hakim (2013)

May be you have already installed R (https://www.r-project.org) and the RStudio (https://www.rstudio.com). you can install the rmarkdown package in R:

# Install from CRAN

install.packages(‘rmarkdown’)

 
 
 


Content for today

April 8, 14:00: Lecture 1 (G. Lohmann)

 

Overview of Dynamics II

Content for today: Fluid Dynamics, Non-dimensional parameters, Dynamical similarity, Elimination of the pressure term and vorticity, bifurcations, applications

 

Basics of Fluid Dynamics

a model describes the state variables, plus fundamental laws and equations of state.

  • State variables: Velocity (in each of three directions), pressure, temperature, salinity, density, etc.
  • Fundamental laws: Conservation of momentum, conservation of mass, conservation of temperature and salinity
  • Equations of state: Relationship of density to temperature, salinity and pressure, etc.

 
 
 


The dynamics of flow are based on the Navier-Stokes equations

\[ \rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \nabla \cdot \mathbb{T} + \mathbf{F}, \]

where \(\mathbf{u}\) is the flow velocity (a vector),

\(\rho\) is the fluid density,

p is the pressure,

\(\mathbb{T}\) is the \(3\times 3\) (deviatoric) stress tensor,

and \(\mathbf{F}\) represents body forces (per unit volume) acting on the fluid.

This equation is often written using the substantive derivative (a statement of Newton’s second law): \[ \rho \frac{D \mathbf{u}}{D t} = -\nabla p + \nabla \cdot\mathbb{T} + \mathbf{F}. \label{newtonzwei} \]

The left side of the equation describes acceleration. The right side of the equation is in effect a summation of body forces (such as gravity) and divergence of stress (pressure and stress).

A very significant feature of the Navier-Stokes equations is the presence of advective acceleration: \[ \mathbf{u} \cdot \nabla \mathbf{u} \]

 

(Script: A general framework can be generally formulated as a transport phenomenon.)

 
 
 


Material laws

The effect of stress represented by \(\nabla p\) and \(\nabla \cdot\mathbb{T}\) terms.

\(\nabla p\) is called the pressure gradient and arises from the isotropic part of the stress tensor.

The anisotropic part of the stress tensor gives rise to \(\nabla \cdot\mathbb{T},\) describing viscous forces.

The stress terms p and \(\mathbb{T}\) are yet unknown: a force model is needed relating the stresses to the fluid motion.

Assumptions on the specific behavior of a fluid are made (based on observations).

The Cauchy stress tensor can be also written in matrix form: \[ \mathbb{T} = \left({\begin{matrix} \mathbf{T}^{(\mathbf{e}_1)} \\ \mathbf{T}^{(\mathbf{e}_2)} \\ \mathbf{T}^{(\mathbf{e}_3)} \\ \end{matrix}}\right) = \left({\begin{matrix} \sigma _{11} & \sigma _{12} & \sigma _{13} \\ \sigma _{21} & \sigma _{22} & \sigma _{23} \\ \sigma _{31} & \sigma _{32} & \sigma _{33} \\ \end{matrix}}\right) \equiv \left({\begin{matrix} \sigma _{xx} & \sigma _{xy} & \sigma _{xz} \\ \sigma _{yx} & \sigma _{yy} & \sigma _{yz} \\ \sigma _{zx} & \sigma _{zy} & \sigma _{zz} \\ \end{matrix}}\right) \equiv \left({\begin{matrix} \sigma _x & \tau _{xy} & \tau _{xz} \\ \tau _{yx} & \sigma _y & \tau _{yz} \\ \tau _{zx} & \tau _{zy} & \sigma _z \\ \end{matrix}}\right) \label{eqn:T} \]

where \(\sigma\) are the normal stresses and \(\tau\) are the shear stresses. From the Newton’s third law (actio est reactio), the stress tensor is symmetrical: \[ \mathbb{T} = \mathbb{T}^T \]

In the Figure the stress vectors \(\mathbf{T}^{(\mathbf{e}_{i})}\) can be decomposed in one normal stress and two shear stress components.

width=3.9 Components of stress in three dimensions.

 
 
 


Conservation of mass

Regardless of the flow assumptions, a statement of the conservation of mass is generally necessary. This is achieved through the mass continuity equation, given in its most general form as: \[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 \] or, using the substantive derivative: \[ \frac{D\rho}{Dt} + \rho (\nabla \cdot \mathbf{u}) = 0. \]

 
 
 


Exercise advection

The temperature at a point 50 km north of a station is 3\(^\circ\)C cooler than at the station. If the wind is blowing from the northeast at 20m/s and the air is being heated by radiation at a rate of 1\(^\circ\)C/h, what is the local temperature change at the station?

 
 
 

Solution of Temperature Advection

The total change of temperature is given by \[ \frac{d T}{dt} = \frac{\partial T}{\partial t} + {\bf u} \cdot \nabla T = \dot{q} \] \[ \Leftrightarrow \quad \frac{\partial T}{\partial t} = - {\bf u} \cdot \nabla T + \dot{q} \] Here we use the velocity \[ {\bf u} = - 20 \frac{m}{s} \cdot \frac{1}{\sqrt{2}} (1, 1, 0) , \qquad \nabla T = \frac{{3}{^\circ C}}{{50}{km}} (0, -1, 0), \qquad \dot{q} = 1 \frac{^\circ C}{h} \] Then we calculate the temperature change at the station \[ \frac{\partial T}{\partial t} = - {\bf u} \cdot \nabla T + \dot{q} \]

\[ \frac{\partial T}{\partial t} = 20 \frac{m}{s} \, \frac{1}{\sqrt{2}} (1, 1, 0) \cdot (0, -1, 0) \frac{{3}{^\circ C}}{{50}{km}} + 1 \frac{^\circ C}{h} \approx {-2.1} \frac{^\circ C}{h} \]

 
 
 


Elimination of the pressure term

in the 2D-Navier-Stokes equation.

2D: assume \(w = 0\) and no dependence of anything on z

\[ \rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}\right) = -\frac{\partial p}{\partial x} + \mu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right) \] \[ \rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y}\right) = -\frac{\partial p}{\partial y} + \mu \left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2}\right) \]

Procedure

  • Differentiating the first with respect to y: \[\partial_y\]

  • the second with respect to x: \[\partial_x\]

  • and subtracting the resulting equations will eliminate pressure and any potential force.

Defining the stream function \(\psi\) through \[ u = \frac{\partial \psi}{\partial y} \quad ; \quad v = -\frac{\partial \psi}{\partial x} \]

results in mass continuity being unconditionally satisfied (given the stream function is continuous), and then incompressible Newtonian 2D momentum and mass conservation degrade into one equation:

\[ \frac{\partial}{\partial t}\left(\nabla^2 \psi\right) + \frac{\partial \psi}{\partial y} \frac{\partial}{\partial x}\left(\nabla^2 \psi\right) - \frac{\partial \psi}{\partial x} \frac{\partial}{\partial y}\left(\nabla^2 \psi\right) = \nu \nabla^4 \psi \]

or using the total derivative

\[ D_t \left(\nabla^2 \psi\right) = \nu \nabla^4 \psi \] \(\nabla^4\) is the (2D) biharmonic operator and \(\nu\) the kinematic viscosity \(\nu=\frac{\mu}{\rho}.\)

This single equation describes 2D fluid flow, kinematic viscosity as parameter!

The concept will become very important in ocean dynamics. The term \[\zeta=\nabla^2 \psi\]

is called relative vorticity, its dynamics can be described as \[ D_t \zeta = \nu \nabla^2 \zeta \quad \]

 
 
 

Non-dimensional parameters: The Reynolds number

For the case of an incompressible flow in the Navier-Stokes equations, assuming the temperature effects are negligible and external forces are neglected.

conservation of mass \[ \nabla \cdot \mathbf{u} = 0 \] conservation of momentum \[ \partial_t \mathbf{u} + ( \mathbf{u} \cdot \nabla) \mathbf{u} = - \frac{1}{\rho_0} \nabla p + \nu \nabla^2 \mathbf{u} \]

The equations can be made dimensionless by a length-scale L, determined by the geometry of the flow, and by a characteristic velocity U.

For analytical solutions, numerical results, and experimental measurements, it is useful to report the results in a dimensionless system (concept of dynamic similarity).

 

Goal: replace physical parameters with dimensionless numbers, which completely determine the dynamical behavior

 

representative values for velocity \((U),\) time \((T),\) distances \((L)\)

Using these values, the values in the dimensionless-system (written with subscript d) can be defined: \[ u =U \cdot u_d \] \[ t = T \cdot t_d \] \[ x = L \cdot x_d \] with \(U = L/T\).

From these scalings, we can also derive \[ \partial_t = \frac{\partial}{\partial t } = \frac{1}{T} \cdot \frac{\partial}{\partial t_d } \] \[ \partial_x = \frac{\partial}{\partial x} = \frac{1}{L} \cdot \frac{\partial}{\partial x_d } \] Note furthermore the units of \([\rho_0] = kg/m^3\), \([p] = kg/(m s^2)\), and \([p]/[\rho_0]= m^2/s^2\). Therefore the pressure gradient term has the scaling \(U^2/L\).

\[ \nabla_d \cdot \mathbf{u_d} = 0 \] and conservation of momentum


  …

The dimensionless parameter \[ Re=UL/ \nu \]

is the Reynolds number and the only parameter left!

For large Reynolds numbers, the flow is turbulent.

In most practical flows \(Re\) is rather large \((10^4-10^8),\) large enough for the flow to be turbulent. A large Reynolds number allows the flow to develop steep gradients locally.

In the literature, the term “equations have been made dimensionless”, means that this procedure is applied and the subscripts d are dropped.

 

Characterising flows by dimensionless numbers

These numbers can be interpreted as follows:

  • Re: (stationary inertial forces)/(viscous forces)
  • Sr: (non-stationary inertial forces)/(stationary inertial forces)
  • Fr: (stationary inertial forces)/(gravity)
  • Fo: (heat conductance)/(non-stationary change in enthalpy)
  • Pe: (convective heat transport)/(heat conductance)
  • Ec: (viscous dissipation)/(convective heat transport)
  • Ma: (velocity)/(speed of sound): objects moving faster than 0.8 produce shockwaves
  • Pr and Nu are related to specific materials.

 
 

Rayleigh-Benard convection (and Rayleigh number)

Bifurcations

Bifurcation youtube, Bifurcation Khan academy, Bifurcation K

Max and Moritz, not lazy, sawing secretly a gap in the bridge.
When now this act is over, you suddenly hear a scream:
“Hey, out! You billy goat!
Tailor, tailor, bitch, bitch, bitch!”
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link

Bridge of Schneider Böck: Stability - Instability

And there he is on the bridge,
Cracks! The bridge is falling apart;

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Linear stability analysis

Consider the continuous dynamical system described by \[ \dot x=f(x,\lambda)\quad \] A bifurcation occurs at \[(x_E,\lambda_0)\] if the Jacobian matrix \[ \textrm{d}f/dx (x_E,\lambda_0)\] has an Eigenvalue with zero real part.

 

Example: transcritical bifurcation

a fixed point interchanges its stability with another fixed point as the control parameter is varied. Bifurcation at \(r=0\).

\[ \frac{dx}{dt}=rx (1-x) \, \]

The two fixed points are 0 and 1. When r is negative, the fixed point at 0 is stable and 1 is unstable. But for \(r>0\), 0 is unstable and 1 is stable.

Stability - Instability: Consumer-producer problem

A typical example could be the consumer-producer problem where the consumption is proportional to the (quantity of) resource.

For example:

\[ \frac{dx}{dt}=rx(1-x)-px \]

where

\[ rx(1-x) \]

is the logistic equation of resource growth:

Rate of reproduction proportional to the population, available resources

\(px\): Consumption, proportional to the resource x.

\[ x_{E 1} = 0, \mbox{ and } x_{E 2 } = 1 - \frac{p}{r} \]

Logistic equation of population growth

Verhulst: describe the self-limiting growth of a biological population with size N:

\[ \frac{dN}{dt}=r N \cdot \left( 1- \frac{N}{K} \right) \]

r growth rate and K carrying capacity.

the early, unimpeded growth rate is modeled by the first term \(r N.\)

“Bottleneck” is modeled by the value of K.

As the population grows, \(-r N^2/K\) becomes large as some members interfere with each other by competing for some critical resource (food, living space). The competition diminishes the combined growth rate, until the value of N ceases to grow (maturity of the population).

\[ N(t) = \frac{K N_0 e^{rt}}{K + N_0 \left( e^{rt} - 1\right)} = \frac{K }{K/N_0 e^{-rt} + 1- e^{-rt} } \quad \rightarrow_{t\to \infty } K \]

In climate, the logistic equation is also important for Lorenz’s forecast error.

Coronavirus epidemic: logistic growth model

N is the number of cases, r infection rate, and K final epidemic size.

dN/dt linearly decreases with the number of cases.

\[ \frac{dN}{dt}=r N \cdot \left( 1- \frac{N}{K} \right) \]

Question: When is the growth rate peak?

How many infections?

Growth rate \(dN/dt\) peak occurs when \(d^2N/dt^2 = 0\) and in time \(t_{pmax} = \ln (K/N_0 - 1)/r\).

At this time the number of cases and the growth rate are

\[ N_{pmax} = K/2 \quad \mbox{and } \quad \frac{ d N (t_{pmax})}{ dt} = rK/4 \]

r=2/150 per day

K=2/3

Human population: logistic growth model

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Explosion of human population over the last 10,000 years along with some relevant historical events.

Think about the ways that each of these events might have affected birth and death rates of the human population.

 
 
 
 
 
 
 

Source: Population dynamics, Nature, 2012

Coronavirus epidemic: logistic growth model

left: 35%

 

Bild Corona
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N is the number of cases,

r infection rate,

p cure rate,

K final epidemic size.

dN/dt linearly decreases with the number of cases.


 
 
 
 
 
 

\[ \frac{dN}{dt}=r N \cdot \left( 1- \frac{N}{K} \right) - p N \]

 
 

Questions:

When is the growth rate peak?

How many infections ?

When do we need places in hospitals?

Coronavirus epidemic: logistic growth model

Using \[ x = N/K \]

\[ \frac{dx}{dt}= f(x) = \underbrace{r x \cdot \left( 1- x \right)}_{growth} \quad - \quad \underbrace{ p x }_{cure} \]

Without any medicine: \[r = 3 p \]

\[ x_{E 1} = 0, \mbox{ and } x_{E 2 } = 1 - \frac{p}{r} = 2/3 \]

Stability: f’(x) at \(x_E\)

\[ f'(x) = r - 2 r x -p \]

\[ f'(x_{E 1})= r-p > 0 \quad \mbox{instability} \]

\[ f'(x_{E 2})= -r+p < 0 \quad \mbox{stability} \]

r=2000/150000

left: 50%

r=2000/150000
Bev=85000000
K=1
dt=0.01
N=150000/Bev
vN=c(0); vNp=c(0); vt=c(0)
vN[1]=N; vNp[1]=0; vt[1]=0

for(i in 2:100000){
N1=N+r*N*(1-N/K)*dt
vNp[i]=r*N*(1-N/K)
vN[i]=N1
vt[i]=i*dt
N=N1}

plot(vt,vN,type="l",xlab="time [days]",ylab="N(t)")

plot(vt,vNp*Bev/100,type="l",xlab="time [days]",ylab="dN(t)/dt  * 1/100  ")

max(vNp[]*Bev/100)

## [1] 2833.333

r=5000/150000

left: 50%

r=5000/150000
Bev=85000000
K=1
dt=0.01
N=10000/Bev
vN=c(0); vNp=c(0); vt=c(0)
vN[1]=N; vNp[1]=0; vt[1]=0

for(i in 2:100000){
N1=N+r*N*(1-N/K)*dt
vNp[i]=r*N*(1-N/K)
vN[i]=N1
vt[i]=i*dt
N=N1}

plot(vt,vN,type="l",xlab="time [days]",ylab="N(t)")

plot(vt,vNp*Bev/100,type="l",xlab="time [days]",ylab="dN(t)/dt  * 1/100  ")

max(vNp[]*Bev/100)

## [1] 7083.333

Saddle-node bifurcation: two fixed points collide

The normal form: \[ \frac{dx}{dt}=r+x^2 \]

\[ r<0: \mbox{ a stable equilibrium point at } -\sqrt{-r} \mbox{ and an unstable one at } \sqrt{-r} \qquad \qquad \]

\(r=0\): exactly one equilibrium point, saddle-node fixed point.

\(r>0\): no equilibrium points. Saddle-node bifurcations may be associated with hysteresis loops.

Saddle-node bifurcation: two fixed points collide

1) Calculate the equilibrium points:

\[ f(x) = r + x^2 = 0 \]

\[ x_{E1,E2} = \pm \sqrt{-r} \mbox{ for } r \le 0 \]

2) Linear stability conditions for the equilibrium points

\[ f'(x) = 2x \]

\[ f'(x_{E1}) = 2\sqrt{-r} > 0 \quad \mbox{unstable} \]

\[ f'(x_{E2}) = -2\sqrt{-r} < 0 \quad \mbox{stable} \]

For r=0: equilibrium point \(x_E=0\) which is indifferent (not stable/unstable).

Saddle-node bifurcation: two fixed points collide

3) Potential or Lyapunov Method

\[ \frac{dx}{dt}=b+x^2 = - \frac{d}{dx} \left( - b x - \frac{x^3}{3} \right) = - \frac{d}{dx} V (x) \]

Global analysis including basins of attraction for \(x_{E2}: (-\infty,3)\)

dev.new(width=5, height=4, unit=“cm”) \[ x_{E1} = 2\sqrt{9} = 6 \quad \mbox{unstable} \]

\[ x_{E1} = -2\sqrt{9} = -6 \quad \mbox{stable} \]

4) Graphical method: slope at equilibrium points

\[ \frac{dx}{dt}=b+x^2 \]

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filled points: positive slope => unstable
open points: negative slope => stable

Convection in the Rayleigh-Benard system

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Rayleigh (1916) temperature difference between the upper- and lower-surfaces \[ T(x, y, z=H) = \, T_0 \] \[ T(x, y, z=0) \, = \, T_0 + \Delta T \]

Furthermore \[ \rho = \rho_0 = const. \] except in the buoyancy term, where:

\[ \varrho = \varrho_0 (1 - \alpha(T-T_0)) \mbox{ with } \alpha > 0 \quad . \]

common feature of geophysical flows

No Convection Equilibrium: Diffusion

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Diffusion: Temperature varies linearly with depth:

\[ T_{eq} = T_0 + \left(1 - \frac{z}{H}\right) \Delta T \]

No movement of particles:

\[ u = w= 0 \]

When this solution becomes unstable, convection should develop.

No Convection Equilibrium: Diffusion

Vorticity in the Rayleigh-Benard system

\[ D_t u = - \frac{1}{\rho_0} \partial_x p + \nu \nabla^2 u \label{eqref:einse} \qquad \qquad \qquad \qquad \qquad \qquad (1) \]

\[ D_t w = - \frac{1}{\rho_0} \partial_z p + \nu \nabla^2 w + g (1- \alpha (T-T_0)) \qquad \qquad (2)\]

\[ \partial_x u + \partial_z w = 0 \qquad \qquad \qquad \qquad \qquad \qquad \qquad (3) \]

\[ D_t T = \kappa \nabla^2 T \qquad \qquad \qquad \qquad \qquad \qquad \qquad (4) \]

 

Introduce vorticity to have (1,2,3) in one equation:

\[ D_t \left( \nabla^2 \Psi\right) = \nu \nabla^4 \Psi - g \alpha \frac{\partial \Theta}{\partial x} \]

Temperature in the Rayleigh-Benard system

\[ T_{eq} = T_0 + \left(1 - \frac{z}{H}\right) \Delta T \]

\[ \mbox{with } \quad T = T_{eq} + \Theta \quad , \quad \mbox{where } \quad \Theta \quad \mbox{is the anomaly} \]

In vorticity equation:

\[ \frac{\partial }{\partial x} g (1- \alpha (T_{eq} + \Theta -T_0)) = - g \alpha \frac{\partial }{\partial x} \Theta \]

Temperature equation:

\[ D_t T = D_t T_{eq} + D_t \Theta = w \cdot \frac{- \Delta T}{H} + D_t \Theta = - \frac{\Delta T}{H} \frac{\partial \Psi}{\partial x} + D_t \Theta \]

\[ D_t \Theta \quad = \frac{\Delta T}{H} \frac{\partial \Psi}{\partial x} + \kappa \nabla^2 \Theta \quad \]

Non-dimensional Rayleigh-Benard system

\[ \frac{1}{T} \frac{1}{L^2} \frac{L^2}{T} D_t \left( \nabla_d^2 \Psi_d\right) = \nu \frac{1}{L^4} \frac{L^2}{T} \nabla_d^4 \Psi_d - g \alpha \frac{ \Delta T}{L} \frac{\partial \Theta_d}{\partial x_d} \]

\[ \frac{\Delta T}{T} D_t \Theta_d \quad = \frac{\Delta T}{H} \frac{L^2}{ T L} \frac{\partial \Psi_d}{\partial x_d} + \kappa \frac{\Delta T}{L^2} \nabla_d^2 \Theta_d \quad \]

This yields (remember \(L=H\))

\[ D_t \left( \nabla_d^2 \Psi_d\right) = \nu \frac{T}{H^2} \nabla_d^4 \Psi_d - g \alpha \frac{T^2 \Delta T}{H} \frac{\partial \Theta_d}{\partial x_d} \]

\[ D_t \Theta_d \quad = \frac{\partial \Psi_d}{\partial x_d} + \kappa \frac{T}{H^2} \nabla_d^2 \Theta_d \quad \]

\[ \mbox{Inserting} \quad T= H^2/\kappa \] \[ \mbox{Rayleigh number} \quad R_a = \frac{g \alpha H^3 \Delta T}{\nu \kappa}\] \[ \mbox{Prandtl number} \quad \sigma = \frac{ \nu}{ \kappa} \]

gives:

\[ D_t \left( \nabla_d^2 \Psi_d\right) = \frac{ \nu}{ \kappa} \nabla_d^4 \Psi_d - g \alpha \frac{H^3 \Delta T}{\kappa^2} \frac{\partial \Theta_d}{\partial x_d} \label{eqref:psieqnn3} \]

\[ D_t \Theta_d \quad = \frac{\partial \Psi_d}{\partial x_d} + \nabla_d^2 \Theta_d \quad . \]

Finally, inserting the non-dimensional parameters

\[ D_t \left( \nabla_d^2 \Psi_d\right) = \sigma \nabla_d^4 \Psi_d - R_a \sigma \frac{\partial \Theta_d}{\partial x_d} \]

\[ D_t \Theta_d \quad = \frac{\partial \Psi_d}{\partial x_d} + \nabla_d^2 \Theta_d \quad \]

Galerkin approximation: Get a low-order model

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\[ \mbox{ Saltzman (1962): Expand } \Psi, \Theta \mbox{ in double Fourier series in x and z: } \]

\[ \Psi (x,z,t) \, = \, \sum_{k=1}^\infty \sum_{l=1}^\infty \Psi_{k,l} (t) \, \, \sin \left(\frac{k \pi a}{H} x \right) \, \times \, \sin \left(\frac{ l \pi}{H} z \right) \] \[ \Theta (x,z,t) \, = \, \sum_{k=1}^\infty \sum_{l=1}^\infty \Theta_{k,l} (t) \cos \left(\frac{k \pi a}{H} x \right) \, \times \, \sin \left( \frac{l \pi}{H} z \right) \]

Approximation: Just 3 Modes X(t), Y(t), Z(t)

\[ \frac{a}{1+a^2} \, \kappa \, \Psi = X \sqrt{2} \sin\left(\frac{\pi a}{H} x \right) \sin\left(\frac{\pi}{H} z \right) \]

\[ \pi \frac{R_a}{R_c} \frac{1}{\Delta T} \, \Theta = Y \sqrt{2} \cos\left(\frac{\pi a}{H} x\right) \sin\left(\frac{\pi}{H} z \right) - Z \sin\left(2 \frac{\pi}{H} z \right) \]

Rayleigh number Ra: Buoyancy & Viscosity

\[ \mbox{Motion develops if } \quad R_a = \frac{g \alpha H^3 \Delta T}{\nu \kappa} \quad \mbox{exceeds } \quad R_c = \pi^4 \frac{(1+a^2)^3}{a^2} \] \[=657.51 \mbox{ occurs when } a^2 = 1/2. \]

\[\mbox{When } R_a < R_c,\mbox{ heat transfer is due to conduction} \]

\[\mbox{When } R_a > R_c, \mbox{ heat transfer is due to convection.} \]

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Lorenz system (also next lecture)

Bifurcation at \[ r = R_a/R_c = 1\]

Geometry constant \[b = 4(1+a^2)^{-1}\]

 
Famous low-order model:

\[ \dot X = -\sigma X + \sigma Y \]

\[ \dot Y = r X - Y - X Z \]

\[ \dot Z = -b Z + X Y \]

 

\[\mbox{dimensionless time } \quad t_d = \pi^2 H^{-2} (1+a^2) \kappa t,\]

\[ \mbox{ Prandtl number } \quad \sigma = \nu \kappa^{-1}, \]

Lorenz system r=24

left: 52%

 

r=24
s=10
b=8/3
dt=0.01
x=0.1
y=0.1
z=0.1
vx<-c(0)
vy<-c(0)
vz<-c(0)
for(i in 1:10000){
x1=x+s*(y-x)*dt
y1=y+(r*x-y-x*z)*dt
z1=z+(x*y-b*z)*dt
vx[i]=x1
vy[i]=y1
vz[i]=z1
x=x1
y=y1
z=z1}
plot(vx,vy,type="l",xlab="x",ylab="y")
plot(vy,vz,type="l",xlab="y",ylab="z")

Lorenz system r=0.9

 

r=0.9
s=10
b=8/3
dt=0.01
x=1.1
y=0.1
z=11.1
vx<-c(0)
vy<-c(0)
vz<-c(0)
for(i in 1:100){
x1=x+s*(y-x)*dt
y1=y+(r*x-y-x*z)*dt
z1=z+(x*y-b*z)*dt
vx[i]=x1
vy[i]=y1
vz[i]=z1
x=x1
y=y1
z=z1}
plot(vx,type="l",xlab="time",ylab="x")
plot(vy,type="l",xlab="time",ylab="y")

Lorenz system r=3.5

 

r=3.5
s=10
b=8/3
dt=0.01
x=1.1
y=0.1
z=11.1
vx<-c(0)
vy<-c(0)
vz<-c(0)
for(i in 1:1000){
x1=x+s*(y-x)*dt
y1=y+(r*x-y-x*z)*dt
z1=z+(x*y-b*z)*dt
vx[i]=x1
vy[i]=y1
vz[i]=z1
x=x1
y=y1
z=z1}
plot(vx,type="l",xlab="time",ylab="x")
plot(vy,type="l",xlab="time",ylab="y")

 

as an outlook: Conveyor Belt

Conveyor
Conveyor

outlook: Thermohaline ocean circulation

Overturning
Overturning

Modelled meridional overturning streamfunction in Sv 10^6 = m^3 /s in the Atlantic Ocean. Grey areas represent zonally integrated smoothed bathymetry