All stresses in this figure have been drawn to be positive. Since the stress tensor is symmetric and is composed by all real numbers, there exist 3 real-valued eigenvalues that we call principal stresses and denote.
Each principal stress eigenvalue is associated with a principal direction eigenvector. Principal directions are always perpendicular to each other in a cartesian coordinate system. When we write the stress tensor in the coordinate system aligned with directions of the principal stresses, the stress tensor results in diagonal elements populated by the principal stresses and zeros in the off-diagonal places.
Usually, the principal stresses are ordered from top to bottom starting with at the top Figure 3. Every tensor with non-zero off-diagonal terms can be simplified to a principal stress tensor with zero off-diagonal terms at the orientation that coincides with the directions of principal stresses.
Consider the schematic in Figure 3. Summation of forces in direction 1, where the term is the body force component, proportional to the solid mass density and volume , and the acceleration component , requires. A generalization of equilibrium in all directions with all stresses Figure 3. In section 2 we will briefly recapitulate the governing equations of motion underlying a simple general circulation model SGCM.
Section 3a introduces a symmetric stress tensor formulation for second-order horizontal diffusion. High-order diffusion schemes are discussed in section 3b.
Section 4 presents SGCM experiments to illustrate the advantages of a symmetric horizontal diffusion scheme. Section 5 gives some concluding remarks. The presentation provides the main formulas while mathematical details can be found in the appendix. Higher orders of horizontal diffusion will be discussed in section 3b.
For high-resolution GCMs it is common to employ 2. This is done to restrict the damping to the smallest resolved scales while leaving the planetary-scale flow almost unaffected.
In this section, several SGCM experiments are considered to explore the consequences of employing nonsymmetric instead of symmetric horizontal diffusion. The present SGCM has already been used in previous studies e. Becker et al. The numerical methods consist of representation by spherical harmonics and finite differences in vertical direction Hoskins and Simmons ; Simmons and Burridge The spatial resolution is spectral truncation at total wavenumber 29 and 32 hybrid layers representing equal mass portions.
In each example sections 4a—c , two runs have been performed using either the symmetric horizontal diffusion scheme 3. In case III, the friction force 3. Such a value is typical for low-resolution climate models e. A thermally balanced zonal flow is a stationary solution of the primitive equations without any friction and diabatic processes.
Even though the present example of a thermally balanced flow is baroclinically unstable, it is conserved by the SGCM as long as wave perturbations are absent. The situation is different if horizontal diffusion is applied while surface drag vertical diffusion and external diabatic heating Q are still neglected. Figure 2 shows the relative temperature changes, the zonal winds, and meridional mass streamfunctions at day for cases I and II.
It is reasonable to calculate a momentary mass streamfunction since the time evolution of the system is very slow and nonoscillatory. Both simulations appear qualitatively equivalent. We observe similar temperature deviations relative to the initial condition Figs. These changes reflect adiabatic heating and cooling associated with the meridional circulations Figs.
At day , both zonal wind profiles Figs. Close to the surface, the zonal winds change sign. Despite these similarities, quantitative differences between both simulations are obvious even from the representation in Fig. Figure 3 shows 4. In comparison with the conventional horizontal diffusion case II, short-dashed lines in Fig.
Note that in case I the increase of TK from about day on does not indicate violation of irreversibility but rather reflects a transfer from available potential energy into kinetic energy via the action of the meridional circulation. Total energy and total angular momentum are well conserved if the horizontal momentum diffusion 3. However, total energy is no more conserved. This result emphasizes the importance of accounting for the frictional heating. The conventional formula 3.
At day , these losses are of the same order of magnitude as the kinetic energy and relative angular momentum of the initial state. The thermally balanced flow of Fig. Figure 4 shows available potential energy due to the zonal-mean flow AZ and the eddies AE as defined by Lorenz together with the corresponding kinetic energies KZ and KE The well-known growth and decay of a baroclinic wave Simmons and Hoskins is evident from each simulation.
The behavior of KZ Fig. Also the zonally averaged temperature and zonal wind at day 40 look similar. Figures 5a,c show corresponding results for case I only. However, computing the differences case II minus case I reveals significant anomalies due to the differences in the horizontal diffusion schemes. The time evolutions of 4.
Total energy and total angular momentum are approximately conserved in case I only solid lines. Total kinetic energy as well as relative and absolute angular momentum behave qualitatively similar in each case I—III.
There is a continuous spurious loss of total angular momentum in case II short—dashed lines. The worst shortcoming of conventional horizontal diffusion consists of a substantial loss of total energy. Within a few days during the life cycle maximum, this loss is of the same magnitude as the overall kinetic energy of the model atmosphere. From the third life cycle experiment case III, long-dashed lines in Fig.
This result may be interpreted as follows. During the phase of baroclinic instability days 0—12 , total kinetic energy grows at the cost of total potential energy. Since the accompanying loss of available potential energy is almost the same, the gain of kinetic energy can be considered as a reversible process.
In other words, enstrophy cascade and thus horizontal diffusion are not very effective. From day 14 on, kinetic energy is lost while available potential energy does not increase again. Hence, the decay phase of a baroclinic life cycle is strongly irreversible due to enstrophy transfer into small scales and the action of horizontal momentum diffusion. However, there must be some diabatic heating to balance the frictional loss of kinetic energy.
Only a symmetric horizontal diffusion scheme completed by the proper dissipation accounts for the corresponding irreversible increase of total potential energy. Climatological temperature, zonal wind and meridional mass streamfunction are shown in Figs.
To illustrate the model response to nonsymmetric horizontal diffusion and neglection of dissipation, Figs. The anomalies are considerably weak. Nevertheless they remain almost the same if the model is integrated for another days, indicating their statistical significance.
A systematic bias in the model climatology can also be deduced from globally integrated properties. As an example, Fig. For conventional horizontal diffusion, both energies are systematically weaker than for the new scheme 3. The overall bias in total potential energy is of the same order of magnitude as the total kinetic energy. Figures 9c,d show the left- and right-hand sides of the energy budget 4. Analogously to the angular momentum budget, the northward flux of total enthalpy does hardly vary among the three cases.
The sum of heating and frictional work adequately balances the enthalpy flux in case I only solid lines. A significant spurious residuum is maintained northward of the winter Hadley cell if the frictional heating rates 2. It has been proven that 2. The central importance of a symmetric stress tensor corresponds to the Eulerian law of angular momentum conservation for arbitrary control volumes.
In addition, energy conservation does crucially depend on the parameterization of frictional processes as well. For a symmetric stress tensor of horizontal diffusion, the local frictional heating rate dissipation is well defined and can easily be accounted for.
The general formula 3. Therefore, the physical meaning of higher orders is open to question even if the stress tensor is symmetric. The physical consistency of a GCM is improved if turbulent friction is based on a symmetric stress tensor. This has been demonstrated by idealized SGCM experiments for the examples of a thermally balanced zonal flow, a baroclinic life cycle, and multiple life cycles.
In case of vanishing external torques and heat sources, conventional horizontal diffusion causes significant losses of total angular momentum and total energy. While the first occurs due to asymmetry of the stress tensor, the latter is due to the lack of frictional heating. Analogous shortcomings can be found in multiple life cycle experiments when evaluating the axiomatic forms of angular momentum and energy conservation for finite control volumes.
A review on nonlinear parameterizations of turbulent viscosities by Smagorinsky also provides a symmetric stress tensor formulation of second-order horizontal momentum diffusion. It differs from the present result 3. After deducing the final formulae for friction force and dissipation with regard to 2. Corresponding results have not been shown since they are almost identical to those obtained using the proposed method.
The conventional biharmonic horizontal diffusion has often been modified in GCM applications such as to conserve a superrotation or to achieve a realistic energy spectrum in longtime simulations.
Nevertheless, spurious torques and heat sinks have hardly been removed from the equations of motion since emphasis has been spent on the formulation of the friction force rather than the stress tensor. In this context it is worth mentioning that also the frictional heating due to vertical momentum diffusion is usually neglected in the design of GCMs.
In this study we have investigated the properties of explicit horizontal diffusion because it is the usual ansatz to account for scale-selective damping in a spectral GCM. Gridpoint models do often involve filter methods instead of an explicit diffusion. Therefore, the compatibility of a particular filter algorithm with the constraints of a symmetric stress tensor, energy conservation, and irreversibility deserves to be investigated. The importance of accounting for first principles in the design of a climate or weather forecast model is evident.
Present SGCM experiments suggest that replacing conventional turbulent friction by the proposed formulation or that of Smagorinsky has weak, but nevertheless significant, effects on the climatology of a GCM. It is thus worthwhile to ask whether delicate properties of comprehensive climate models, like internal variability or the response to a perturbation in the external forcing, are possibly affected by the intrinsic torques and cooling rates associated with conventional formulations of turbulent friction.
Encouraging comments and suggestions by G. Schmitz and one anonymous reviewer have greatly improved the manuscript. Also the valuable report of a second anonymous reviewer is gratefully acknowledged. I will answer these one at a time.
Both answers will be done by analogy with the fluid flow. Concerning the symmetry: if the stress tensor were not symmetric there would be a net torque on the object and it would rotate.
For an explanation see the image below. One important thing to remember is that there is the supposition that there is no force that acts on the bulk of the object that's capable of exert net torque i. This idea changes when you think on something like the interplay between matter and electromagnetic field. Although you can adapt the image above to show that the full stress tensor of the whole system must be symmetric, it's not necessary the stress tensor of it's part should be symmetric too.
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