physics

Preparing for the assimilation of future ocean-current measurements

The Social MetworkLeave a comment

By Laura Risley

Ocean data assimilation (DA) is vital. Firstly, it is essential to improving forecasts of ocean variables. Not only that, the interaction between the ocean and atmosphere is key to numerical weather prediction (NWP) as coupled ocean-atmosphere DA schemes are used operationally.  

At present, observations of the ocean currents are not assimilated operationally. This is all set to change, as satellites are being proposed to measure these ocean currents directly. Unfortunately, the operational DA systems are not yet equipped to handle these observations due to some of the assumptions made about the velocities. In my work, we propose the use of alternative velocity variables to prepare for these future ocean current measurements. These will reduce the number of assumptions made about the velocities and is expected to improve the NWP forecasts.

What is DA? 

DA combines observations and a numerical model to give a best estimate of the state of our system – which we call our analysis. This will lead to a better forecast. To quote my lunchtime seminar ‘Everything is better with DA!’. 

Our model state usually comes from a prior estimate which we refer to as the background. A key component of data assimilation is that the errors present in both sets of data are taken into consideration. These uncertainties are represented by covariance matrices. 

I am particularly interested in variational data assimilation, which formulates the DA problem into a least squares problem. Within variational data assimilation the analysis is performed with a set of variables that differ from the original model variables, called the control variables. After the analysis is found in this new control space, there is a transformation back to the model space. What is the purpose of this transformation? The control variables are chosen such that they can be assumed approximately uncorrelated, reducing the complexity of the data assimilation problem.

Velocity variables in the ocean 

My work is focused on the treatment of the velocities in NEMOVAR. This is the data assimilation software used by the NEMO ocean model, used operationally at the Met Office and ECMWF. In NEMOVAR the velocities are transformed to their unbalanced components, and these are then used as control variables. The unbalanced components of the velocities are highly correlated, therefore contradicting the assumption made about control variables. This would result in suboptimal assimilation of future surface current measurements – therefore we seek alternative velocity control variables. 

The alternative velocity control variables we propose for NEMOVAR are unbalanced streamfunction and velocity potential. This would involve transforming the current control variables, the unbalanced velocities, to these alternative variables using Helmholtz Theorem. This splits a velocity field into its nondivergent (streamfunction) and irrotational (velocity potential) parts. These parts have been suggested by Daley (1993) as more suitable control variables than the velocities themselves. 

Numerical Implications of alternative variables 

We have performed the transformation to these proposed control variables using the shallow water equations (SWEs) on a 𝛽-plane. To do so we discretised the variables on the Arakawa-C grid. The traditional placement of streamfunction on this grid causes issues with the boundary conditions. Therefore, Li et al. (2006) proposed placing streamfunction in the centre of the grid, as shown in Figure 1. This circumvents the need to impose explicit boundary conditions on streamfunction. However, using this grid configuration leads to numerical issues when transforming from the unbalanced velocities to unbalanced streamfunction and velocity potential. We have analysed these theoretically and here we show some numerical results.

Figure 1: The left figure shows the traditional Arakawa-C configuration (Lynch (1989), Watterson (2001)) whereby streamfunction is in the corner of each grid cell. The right figure shows the Arakawa-C configuration proposed by Li et al. (2006) where streamfunction is in the centre of the grid cell. The green shaded region represents land. 

Issue 1: The checkerboard effect 

The transformation from the unbalanced velocities to unbalanced streamfunction and velocity potential involves averaging derivatives, due to the location of streamfunction in the grid cell. This process causes a checkerboard effect – whereby we have numerical noise entering the variable fields due to a loss of information. This is clear to see numerically using the SWEs. We use the shallow water model to generate a velocity field. This is transformed to its unbalanced components and then to unbalanced streamfunction and velocity potential. Using Helmholtz Theorem, the unbalanced velocities are reconstructed. Figure 2 shows the checkboard effect clearly in the velocity error.

Figure 2: The difference between the original ageostrophic velocity increments, calculated using the SWEs, and the reconstructed ageostrophic velocity increments. These are reconstructed using Helmholtz Theorem, from the ageostrophic streamfunction and velocity potential increments. On the left we have the zonal velocity increment error and on the right the meridional velocity increment error. 

Issue 2: Challenges in satisfying the Helmholtz Theorem 

Helmholtz theorem splits the velocity into its nondivergent and irrotational components. We discovered that although streamfunction should be nondivergent and velocity potential should be irrotational, this is not the case at the boundaries, as can be seen in figure 3. This implies the proposed control variables are able to influence each other on the boundary. This would lead to them being strongly coupled and therefore correlated near the boundaries. This directly conflicts the assumption made that our control variables are uncorrelated. 

Figure 3: Issues with Helmholtz Theorem near the boundaries. The left shows the divergence of the velocity field generated by streamfunction. The right shows the vorticity of the velocity field generated by velocity potential. 

Overall, in my work we propose the use of alternative velocity control variables in NEMOVAR, namely unbalanced streamfunction and velocity potential. The use of these variables however leads to several numerical issues that we have identified and discussed. A paper on this work is in preparation, where we discuss some of the potential solutions. Our next work will further this investigation to a more complex domain and assess our proposed control variables in assimilation experiments. 

References: 

Daley, R. (1993) Atmospheric data analysis. No. 2. Cambridge university press. 

Li, Z., Chao, Y. and McWilliams, J. C. (2006) Computation of the streamfunction and velocity potential for limited and irregular domains. Monthly weather review, 134, 3384–3394. 

Lynch, P. (1989) Partitioning the wind in a limited domain. Monthly weather review, 117, 1492–1500. 

Watterson, I. (2001) Decomposition of global ocean currents using a simple iterative method. Journal of Atmospheric and Oceanic Technology, 18, 691–703

How does plasma from the solar wind enter Earth’s magnetosphere?

Sarah Bentley1 Comment

Earth’s radiation belts are a hazardous environment for the satellites underpinning our everyday life. The behaviour of these high-energy particles, trapped by Earth’s magnetic field, is partly determined by the existence of plasma waves. These waves provide the mechanisms by which energy and momentum are transferred and particle populations physically moved around, and it’s some of these waves that I study in my PhD.

However, I’ve noticed that whenever I talk about my work, I rarely talk about where this plasma comes from. In schools it’s often taught that space is a vacuum, and while it is closer to a vacuum than anything we can make on Earth, there are enough particles to make it a dangerous environment. A significant amount of particles do escape from Earth’s ionosphere into the magnetosphere but in this post I’ll focus on material entering from the solar wind. This constant outflow of hot particles from the Sun is a plasma, a fluid where enough of the particles are ionised that the behaviour of the fluid is then dominated by electric and magnetic fields. Since the charged particles in a plasma interact with each other, with external electric and magnetic fields, and also generate more fields by moving and interacting, this makes for some weird and wonderful behaviour.

magnetosphere_diagram
Figure 1: The area of space dominated by Earth’s magnetic field (the magnetosphere) is shaped by the constant flow of the solar wind (a plasma predominantly composed of protons, electrons and alpha particles). Plasma inside the magnetosphere collects in specific areas; the radiation belts are particularly of interest as particles there pose a danger to satellites. Credit: NASA/Goddard/Aaron Kaas

When explaining my work to family or friends, I often describe Earth’s magnetic field as a shield to the solar wind. Because the solar wind is well ionised, it is highly conductive, and this means that approximately, the magnetic field is “frozen in” to the plasma. If the magnetic field changes, the plasma follows this change. Similarly, if the plasma flows somewhere, the magnetic field is dragged along with it. (This is known as Alfvén’s frozen in theorem – the amount of plasma in a volume parallel to the magnetic field line remains constant). And this is why the magnetosphere acts as shield to all this energy streaming out of the Sun – while the magnetic field embedded in the solar wind is topologically distinct from the magnetic field of the Earth, there is no plasma transfer across magnetic field lines, and it streams past our planet (although this dynamic pressure still compresses the plasma of the magnetosphere, giving it that typical asymmetric shape in Figure 1).

Of course, the question still remains of how the solar wind plasma enters the Earth’s magnetic field if such a shielding effect exists. You may have noticed in Figure 1 that there are gaps in the shield that the Earth’s dipole magnetic field presents to the solar wind; these are called the cusps, and at these locations the magnetic field connects to the solar wind. Here, plasma can travel along magnetic field lines and impact us on Earth.

But there’s also a more interesting phenomenon occurring – on a small enough scale (i.e. the very thin boundaries between two magnetic domains) the assumptions behind the frozen-in theorem break down, and then we start to see one of the processes that make the magnetosphere such a complex, fascinating and dynamic system to study. Say we have two regions of plasma with opposing orientation of the magnetic field. Then in a middle area these opposing field lines will suddenly snap to a new configuration, allowing them to peel off and away from this tightly packed central region. Figure 2 illustrates this process – you can see that after pushing red and blue field lines together, they suddenly jump to a new configuration. As well as changing the topology of the magnetic field, the plasma at the centre is energised and accelerated, shooting off along the magnetic field lines. Of course even this is a simplification; the whole process is somewhat more messy in reality and I for one don’t really understand how the field can suddenly “snap” to a new configuration.

reconnection
Figure 2: Magnetic reconnection. Two magnetic domains of opposing orientation can undergo a process where the field line configuration suddenly resets. Instead of two distinct magnetic domains, some field lines are suddenly connected to both, and shoot outwards and away, as does the energised plasma.

In the Earth’s magnetosphere there are two main regions where this process is important (Figure 3). Firstly, at the nose of the magnetosphere. The dynamic pressure of the solar wind is compressing the solar wind plasma against the magnetospheric plasma, and when the interplanetary magnetic field is orientated downwards (i.e. opposite to the Earth’s dipole – about half the time) this reconnection can happen. At this point field lines that were solely connected to the Earth or in the solar wind are now connected to both, and plasma can flow along them.

magnetosphere_reconnection_sites
Figure 3: There are two main areas where reconnection happens in Earth’s magnetosphere. Opposing field lines can reconnect, allowing a continual dynamic cycle (the Dungey cycle) of field lines around the magnetosphere. Plasma can travel along these magnetic field lines freely. Credits: NASA/MMS (image) and NASA/Goddard Space Flight Center- Conceptual Image Lab (video)

Then, as the solar wind continues to rush outwards from the Sun, it drags these field lines along with it, past the Earth and into the tail of the magnetosphere. Eventually the build-up of these field lines reaches a critical point in the tail, and boom! Reconnection happens once more. You get a blast of energised plasma shooting along the magnetic field (this gives us the aurora) and the topology has rearranged to separate the magnetic fields of the Earth and solar wind; once more, they are distinct. These dipole field lines move around to the front of the Earth again, to begin this dramatic cycle once more.

Working out when and how these kind of processes take place is still an active area of research, let alone understanding exactly what we expect this new plasma to do when it arrives. If it doesn’t give us a beautiful show of the aurora, will it bounce around the radiation belts, trapped in the stronger magnetic fields near the Earth? Or if it’s not so high energy as that, will it settle in the cooler plasmasphere, to rotate with the Earth and be shaped as the magnetic field is distorted by solar wind variations? Right now I look out my window at a peaceful sunny day and find it incredible that such complicated and dynamic processes are continually happening so (relatively) nearby. It certainly makes space physics an interesting area of research.