# Chapter 9

Perturbation by Noise

In this chapter, we take up the problem of the climate system's energy balance disturbed by noise. The noise term is taken to imitate weather and other small space–time scale perturbations of the energy balance. First, we consider the basic case of the model with constant coefficients on the sphere. This starting point is in line with our approach throughout the book of step-by-step understanding of the convergence toward a more comprehensive and realistic climate model. The new element in our process is an attempt to capture the fluctuations of the system about the ensemble mean. The idea was introduced in Chapter 2 for the global average model. In Section 6.5, we introduced the idea of fluctuations in the system due to random winds (departures from normal mean circulations) whose timescales are shorter than the relaxation timescale of a column of air. There are other physical elements besides the winds, which fluctuate with such short timescales, for example, areal extent and height of cumulus clouds, passage of mid-latitude weather systems, water vapor concentration in three dimensions, and aerosol particle concentrations. Some of these can be lumped on the driver's side of the EBM governing equation, while the effects of horizontal heat transport fluctuations are located in the advection term (divergence of heat flux). If we make the assumption that the heat advection term can be decomposed,

**9.1**

where the angle brackets mean ensemble averaging (or expectation value in the probabilistic sense), and the energy balance climate model (EBCM) surface temperature in all the previous chapters is now , and is a random field representing timescales much shorter than the relaxation time, , and spatial scales less than those of a spherical harmonic of degree 11. These latter would include all the fluctuations we might associate with “weather.”

In most of the treatment in this chapter, we deal with anomalies, that is, the departures from the ensemble mean at a given point on the sphere, .

**9.2**

We drop the prime hereafter and refer simply to as the anomaly.

Our next task (as usual) is to solve some models with ideal geography and then proceed to more complicated ones. The models will have the governing equation we are accustomed to, except for the noise driver, on the RHS. We will be seeking statistical quantities from the solutions to problems such as for a given weather noise forcing, what is the response in terms of the distribution of variance in the response modes.

## 9.1 Time-Independent Case for a Uniform Planet

In this section we first consider an imaginary Earth that is spatially uniform with respect to the time-independent problem dealt with earlier but forced by spatially dependent noise:

where is spatially white noise and that means it satisfies

where . This kind of random field evaluated at one point is *uncorrelated* with the field evaluated at another point even for very tiny separation distances. In our problem, it means the variability is in the form of eddies in space that are small compared to the natural length scales in the problem; basically, the natural length scales are much smaller than , expressed in units of the Earth's radius.

We say the forcing is *spatially white noise*. We can expand the white noise random forcing field into a Laplace series (i.e., into spherical harmonic components as in Chapter 8). Note that from here we drop the subscript “noise” to keep the notation simpler.

where the Laplace series components are complex random numbers which we will take to be normally distributed. Using the orthogonality of the spherical harmonics, we obtain the inverse to be

**9.6**

Consider the covariance between these components with different indices. Using the properties, we find that

**9.7**

Next substitute the expression for the quantity in angular brackets in (9.4) and the resulting expression reduces to

**9.8**

This last equation tells us a lot about the covariances between the Laplace components of spatially white noise. The covariance vanishes unless and . There are no cross-covariances! Each component is statistically independent of every other component. In addition, the variance associated with each component , is the same for every component indexed. For each spherical harmonic degree, , there are , components, each of which has the same variance.

White noise is a special case of the more general condition of statistically rotationally invariant random fields on the sphere. When ensemble averages of the mean and some second moments of a random field are rotationally invariant on the sphere, it is possible to decompose the variance of the random field into a spectrum of variances analogous to the treatment of a stationary time series where the symmetry or invariance is along the timeline. We will find that in some cases this can be utilized on the uniform-sphere models to follow.

Returning to the white noise spatial process we will use to perturb the energy balance, it is helpful to think of *realizations* of the white noise field . We can generate a realization of the field by first going to a complex Gaussian random number generator and pulling out one random number whose variance is (square of the sum of the real and imaginary parts of ) for , then repeating this to draw statistically independent values for each with exactly the same variance , and so on. We then take these complex random numbers and enter them in the formula for given by (9.5). Now turn to the solution for the temperature components. We can expand

**9.9**

and its inverse,

**9.10**

By substituting and using the fact that the are the eigenfunctions of with eigenvalue and then using the orthogonality of the , we have

with , where we have taken the Earth's radius to be unity. The are complex random numbers and the randomness comes from the factor . The factor within the large parentheses weights the () components of according to the dependence in the denominator. Note that the proportionality factor does not contain any dependence.

The covariance between different components can be readily calculated:

**9.12**

with

**9.13**

Note that there is no dependence. There are two important parameters: and . The first governs the overall variance of , the second determines how the variance from the white noise forcing is apportioned by the proportionality factor in (9.11), or more physically the inverse of the operator (see 9.3) which operates on . The factor in the large parentheses can be thought of as a “filter” that modifies the input variance, . This filter allows low modes (larger scales) to pass from the stimulus to the corresponding modes of the response but reduces the power (or variance) passed to the higher index modes. This filter defines the dynamical character of the damped diffusion operator. The damped diffusion filter smooths out the highly erratic (high-mode-index) white noise input.

The covariance of the temperature field is given by

**9.14**

After making use of the Kronecker deltas, we have

**9.15**

Now we can employ a wonderful theorem called the *addition theorem for spherical harmonics*.^{1} The theorem states

**9.16**

Inserting this result, we find that

**9.17**

This last formula tells us that the covariance of the temperature field with itself depends only on the opening angle from the Earth's center between the two points at the surface. (remember here). The opening angle is simply . This is the condition for rotational invariance on the sphere. It is hardly a surprise for the case with constant coefficients, as the operator (being a scalar product ) is also rotationally invariant along with all the other terms on the sphere. Figure 9.1a shows an example of a degree spectrum of the white noise variance as a function of degree . The spectrum increases as a function of because there are modes for each value of . Figure 9.1b shows the responding temperature field utilizing a value of . Note that the degree spectrum begins to turn over at and the growing terms in the denominator begin to dominate and filter out the high wavenumber stimulus of the white spatial noise in the numerator.

We cannot ignore the opportunity to draw a parallel with empirical orthogonal functions (EOFs), which have the property that if a random field is expanded into these orthogonal functions, the expansion coefficients are statistically independent. We have just proved (with the help of the spherical harmonic addition theorem) that the are the EOFs of any random field whose statistics are rotationally invariant on the sphere. There is nothing empirical here, so a comment is called for. Karhunen and Loève studied the basis sets of random fields, not just generated from data but from theoretically generated continuous random fields. The basis sets that do it are now called the *Karhunen–Loève* functions. When we use EOFs to reduce the dimension of an empirical “random” field we call the basis set the *EOFs*. The EOFs are just the eigenvectors of the cross-covariance matrix. For the case of the sphere, the Karhunen–Loève functions are the eigenfunctions of the kernel :

**9.18**

where is the eigenvalue and is proportional to the variance of the particular EOF coefficient labeled . This result can be proved by expanding into Legendre polynomials and using the additiontheorem.^{2} We have the remarkable coincidence (rotational invariance) in our climate model (with uniform properties) that the KL functions are also the dynamical normal modes—or in our damped diffusive problem, the decay modes. This happens in the present case because the rotational symmetry forces it. The variances associated with the modes of also do not depend on the index , which is a consequence of the rotational invariance as well.

We turn to the dependence of (note: there is no dependence):

**9.19**

The *degree variance* which includes the variance contributed by longitudinal modes is

**9.20**

which is shown in Figure 9.1b for . For reference, Figure 9.1a shows the degree spectrum of the white noise driving force. If were vanishing, the degree spectrum would expand into the white noise spectrum in Figure 9.1a. Instead, contributions from the upper part of the variance index are shut out by the damped-diffusive operator for . We call operators like this *low-pass filters*. In other words, only the low-index modes from the forcing are passed to the response spectrum of variances.

## 9.2 Time-Dependent Noise Forcing for a Uniform Planet

Now we admit the heat storage term to the energy balance equation to permit time dependence in the problem. We continue with constant for a uniform planet. The noise agent forcing the system will also be white in time as well as in space, keeping the conditions for rotational invariance on the sphere as well as the conditions of stationarity for the time series representing the temperature field. The spatial white noise is quite different from that of planet Earth as, in reality, the conditions for weather noise would not be rotationally invariant on our planet. Weather noise intensity (variance) is seasonally dependent and is restricted to a seasonally cycling band in the middle latitudes of eachhemisphere.

We prescribe the noise to be white in time as well as in space. This means the noise has a very short autocorrelation time (equal to its relaxation time, e.g., over land, days) compared to that of actual weather whose characteristic time is a few days. We might think of the noise here as the fluctuations of middle-latitude weather, which has an autocorrelation timescale of about 3 days. The mean of the noise function vanishes:

**9.21**

and its autocovariance function in space–time is given by

**9.22**

We must allow the forcing and response (random) fields to be decomposed in frequency as well as spherical harmonic components. This will be possible as the model forcing as well as the solution will be a stationary and continuous time series. The time series span is . Since the time is continuous (as opposed to jumping in discrete steps), we are obliged to use the continuous Fourier transform.

**9.23**

**9.24**

where we have used the superscript to denote the Fourier component corresponding to frequency . We also employ a tilde, to denote the Fourier-transformed variable as opposed to the Fourier mirror image, . By substituting and using the fact that the are the eigenfunctions of and then using the orthogonality of the , we have

**9.25**

and we have, in addition,

**9.26**

Because the time is continuous, we need the Dirac delta function, in the autocovariance of white noise. Just as before, we can write the relations, but now including frequency:

**9.27**

with

**9.28**

where

**9.29**

We can now compute the autocovariance of the temperature field evaluated at two separated points on the sphere and and at two times and . The separations are lag and , the latter being the great circle distance from the two points on the unit sphere. We start with

**9.30**

After use of the techniques above, including the addition theorem, we obtain for :

Note that the result does not depend on (all points on this sphere are the same). We have one more integral to deal with (we can turn to tables or MATHEMATICA):

**9.32**

The result to be used in (9.31) is a sum of exponentially decaying terms. The solid curve in Figure 9.2 shows a log-plot of the sum , using , and retaining 10 terms (more terms do not affect the result). The solid curve is the sum, and the dashed curve is the leading term in the sum.

## 9.3 Green's Function on the Sphere:

Consider the response of the temperature field to a steady heat source located at the point . We examine first the time-independent case

**9.33**

where denotes the thermal response field to the point source. The function is called *Green's function* for the field. It can be seen by symmetry that it has the property that its value depends only on the great circle distance from the source . It is especially of interest as the thermal field response to an arbitrary distribution of heat, say, leads to a thermal anomaly and that anomaly can be related to the Green's function in a relatively simple way. According to our definitions,

**9.34**

Now expand all the functions into their Laplace series:

**9.35**

**9.36**

**9.37**

**9.38**

The expansion of comes about by first expanding into Legendre polynomials . The others are conventional expansions that can be checked from definitions.

By the usual insertion and projection of spherical harmonic components,

**9.39**

**9.40**

Now consider the thermal anomaly

**9.41**

The last, which is our desired result, can be obtained by inserting the Laplace series for the factors in the integrand.

## 9.4 Apportionment of Variance at a Point

Next consider the fluctuations of the surface temperature at a point. These fluctuations may be thought of as being composed of contributions from all space and timescales. For example, consider the uniform earth case ( and are constant). The Fourier component corresponding to frequency of the temperature at point is given by

**9.42**

The variance at point and frequency is given by

**9.43**

The total variance at degree (including contributions for all modes of the degree level) is

**9.44**

Figure 9.3 shows the fraction of total variance contained in a given spherical harmonic of degree for and as a function of . As the frequency is increased from to , the power moves to higher-degree indices. This figure is to be compared with Figure 9.1 where the cases for and the case for the integral over all frequencies (equivalent to the distribution of variance at a point in time).

## 9.5 Stochastic Model with Realistic Geography

As in the last chapter, models with a realistic land–sea distribution cannot be solved analytically. One must turn to numerical methods to obtain solutions. The introduction of noise as a forcing agent is not too difficult. First of all, the EBCM is basically a linear system (if we ignore snow and other nonlinear feedbacks) with time-independent coefficients. This means that the model that simulates the seasonal cycle (Chapter 8) can be used by simply removing the seasonal driving term and inserting the noise field.

Leung and North (1991) compared a general circulation model (GCM) simulating climate on a bald, land-covered planet (referred to in Chapter 1 as *Terra Blanda*) run at equinox conditions with models of the all land models of the type studied earlier in this chapter. The two modeling schemes had similar spatial statistics. As an example, consider the relaxation time for Legendre modes as shown in Figure 9.4. The fit to the relaxation times in GCM and EBCM for a bald planet is remarkable. Note that the GCM runs were set at equinox conditions and the EBCM is set at mean annual values (in the linear ECBM, this does not matter!). Moreover, the GCM statistics are hardly rotationally invariant on the sphere. Because of the featureless geography, its solution statistics are longitudinally stationary, but not the latitudinal ones. The results of this study by Leung and North together with the seasonal modeling success of the full EBM suggests that we might have a chance at modeling the statistics of the response to white space–time noise forcing.

The linear 2-D EBCM of North *et al*. (1983), which used the spherical harmonic basis, and many later versions of it have been solved by various methods including finite differencing on the spherical surface (Wu and North, 2007), finite differencing employing multigrid relaxation (Bowman and Huang, 1991; Stevens and North, 1996). A novel means of solving the model with geography involves using the nonorthogonal decay modes as a basis set. The first study to attempt including noise with the current land–sea distribution, this was carried out by Kim and North (1991), who employed the spherical harmonic basis set (Wu and North, 2007).

We take figures here from the two-dimensional EBCM of Kim and North (1991). This model employed a simple mixed-layer ocean and the forcing noise was white in space and time. Note especially that the space–time white noise was uniformly distributed over the globe in variance and with zero mean. Some later model studies attempted to apportion the forcing noise to be only in the mid-latitudes, but here we show only the uniformly distributed case, as otherwise we would have to introduce more phenomenological parameters. The only adjustable parameter is then the variance of the noise field, . The variance of the temperature field is everywhere proportional to this variance. We adjust this variance to match the observed variance to that of the model's simulated temperature fluctuations. Note that in all cases in this chapter, the noise-forced EBCM has a mixed-layer (slab) ocean. Low-frequency variability would be somewhat different in a model with deeper oceanic elements.

Figures 9.5–9.7 show a sequence of maps of the variance of the surface temperature field in the observations. The left panels indicate observations and the right panels are for EBCM simulations. Both fields were smoothed to the same level of T11. The data and observations were band-pass filtered to include the periods between the limits indicated on the maps. Procedures for the filtering are found in the paper by Kim and North (1991). It is interesting that once the model's free parameters are adjusted (tuned) to match the variance of the observations in the middle of Asia, the maps come into reasonable agreement elsewhere, for example 2 C in Antarctica and 0.2 C in South America (Figure 9.5). As in the seasonal model, the response to high-frequency forcing is dominated by the land–sea configuration. Note also the influence of the Himalayan plateau on the observations, but not on the simulated field, which of course has no topography. Topography is likely to be at work also in northwestern North America. The agreement in this procedure emphasizes the dominant nature of the geographical imprint of the land–sea distribution of the surface temperature.

Figure 9.6 is the same as the last figure except that the frequencies lie between periods of 1 and 10 years. In this figure, we also note the strong effects of the Himalayas and also a hint of the ENSO pattern in the data, but not, of course, in the mixed-layer ocean model. Figure 9.7 showing fluctuations having periods between 10 and 30 years shows some strong indications in the data of sea ice fluctuations around the northern edges of the continents. There is also a hint of ENSO in the data, but most ENSO is at higher frequencies than this band permits. Note the overall washing out of the features of the EBCM simulation in these low-frequency components. A deeper ocean than the mixed layer shown here would show more continental–oceanic contrast.

Figures 9.8 and 9.9 show the spatial correlation between six fixed points and their surrounding areas (mid-Asia, Africa, South Pole, mid-Atlantic, equatorial Pacific, West coast-America). In each case, the correlation decays from the fixed point out to where it falls off to 1 indicated by the heavy line. Features to note are that at high frequencies (Figure 9.8), the length scales are long over land and short over ocean. Since the ocean has a timescale of a few years, we are in the high-frequency regime of the oceans in this band. On the other hand, we are in the low-frequency band for land areas and the autocorrelation distances are larger. This is shown dramatically for the fixed point at San Francisco where the correlation out to sea is short and inland it is long. The effect is prominent in both data and model simulation. Note that ENSO is prominent in the data, but missing completely as expected in the EBCM simulation.

The situation in Figure 9.9 is quite different. The correlation lengths over the North Atlantic are too large in the EBCM simulation because of the short timescale of the mixed-layer model. ENSO dominates all of the tropics in the data, but is absent in the EBCM. Our last comparison is for the one-month lagged correlation map between the observations (Figure 9.10a) and the EBCM (Figure 9.10b). In this comparison, we do find some serious discrepancies although the lags over continental interiors is pretty good. The transition from land to oceans is quite abrupt in the EBCM upto the level of about 0.6, while in the data it is smoother to roughly this same value. Also shown in the paper by Kim *et al*. (1996) are comparisons of the same statistics as in Figures 9.6–9.10. (Figures modified from Kim *et al*. (1996). (© Amer. Meteorol. Soc., with permission.))

## 9.6 Thermal Decay Modes with Geography

In this section,^{3} we consider an alternative modal decomposition of the time-dependent problem with real land–sea geography. The model is linear. We begin with the equation for departures of the local surface temperature from steady state:

**9.45**

Next, insert the exponential time dependence . This leads to the eigenvalue problem

**9.46**

with the boundary conditions that the solution be finite and without divergence at the poles. This is called a *generalized Stürm–Liouville system* because of the space-dependent factor on the RHS. Under certain conditions, this system yields a set of eigenfunctions corresponding to eigenvalues , (see Horn and Johnson, 1985). The main conditions are that the domain be finite (the spherical surface, in this case) and that the operator on the LHS of the last equation be Hermitian. An operator in this context is one in which

**9.47**

which can be demonstrated by using integration by parts or equivalently the two-dimensional divergence theorem on the spherical surface. Note that the inverse of the eigenvalues are just the relaxation times for the modes, .

The modes are *not* orthogonal. Nevertheless, we can form series representations because of the following relation (which can be derived from theabove expressions):

where the *inner product notation* is introduced as a notational simplification. These steps lead us to

**9.49**

with

We can now insert (9.57) into the governing equation to find

Note the absence of in the integral on the RHS compared to the previous equation. Here we can see explicitly that if is set to zero the amplitude of mode decays exponentially with time constant .

Returning to the eigenvalue relation, we can use values from the map of in (9.48). The heat capacity map is represented by 64 (longitude) 31 ((latitude)) plus 2 polar grid points. Once the heat capacity matrix is formed, one can use standard methods on (9.48) to recover the eigenvalues, , and the eigenvectors, . Hereafter, we will refer to the as the *thermal decay modes* (*TDMs*). We will now examine a few of the results. First, consider the spectrum of relaxation times, , as shown in Figure 9.11. The dotted curve shows the values that would be obtained from observational data projected onto the modal shapes. The essential difference between these modes and the EOF modes (Kim and North, 1992) is that they are spatial physical modes and their shapes do not depend on frequency as the statistical (EOF) modes do. Also,they are not strictly orthogonal as are the EOFs. ( is the weighting function).

Consider the log–log spectra of relaxation times shown in Figure 9.11. The number of spectral components is 1986, equal to the number of grid points. The spectrum is ordered from the longest time at to the shortest at . The dashed curve shows a smoothed estimate calculated from a model based on the parameterization of Hyde *et al*. (1990). It is interesting that there is a sharp fall in the relaxation times at . This represents the transition from the family of oceanic modes with long timescales to land-dominated modes for . The dotted curve in Figure 9.11 is an estimate of the relaxation time spectrum data. This is done by projecting the observed data onto the model-calculated eigenmode and calculating the autocorrelation times from the resulting time series for each decay mode. Note that the “observed” spectrum is much flatter than the model-generated spectrum (dashed line). One can adjust the parameter values in as well as to bring the spectra more into line. Numerical experimentation shows that the shapes of the decay modes are not significantly affected by a fairly wide range of choices for these parameters. Given the problem with matching the model to data with respect to one-month lags in Figure 9.10, it is not surprising that we might need to make adjustments. More details are contained in Wu and North (2007).

When we examine the oceanic modes (), we find that virtually all of the non-negligible amplitudes are over ocean as shown in Figure 9.12. The lowest modes shown there have time constants between 0.763 and 0.735 year. Since length scales are short over this mixed-layer ocean, these response modes as driven by white noise are virtually white in space as well (no correlation between one point and another on this sparse grid). Next, turn to the land modes, . These modes group themselves into families according to continental clusters. For example, Figure 9.13 shows four modes. Figure 9.13a corresponds to year. Times for the other panels (Figure 9.13b–d) are listed in the figure caption. These modes were selected because they form a family of modes connected with the Eurasian continent. Figure 9.14 shows four modes associated with the North American continent. The modal shapes in both figures show the familiar “drumhead” patterns of eigenmodes for either wavelike or diffusive-like systems.

### 9.6.1 Statistical Properties of TDMs

We have already remarked that the TDMs cannot correspond to EOFs because EOFs are mutually orthogonal, whereas the TDMs are skewed. If the mixed layer were infinitely deep, there would be no response to space–time white noise over the oceans and this would force the TDMs in that limit to be mutually orthogonal.^{4} Looking at the last two figures, one is tempted to think along these lines at least forheuristic purposes.

To get an idea about the statistical properties of TDMs refer to (9.50) and (9.51). First, note that the TDMs are dynamical modes. This means that if certain spatial mode patterns are present in the forcing, only those mode patterns will be found in the response. Similarly, if certain temporal frequencies are present in the forcing, only those frequencies will be found in the response. This follows from the stationarity of the forcing and its response . But the nonorthogonality of the solutions leads to a very peculiar property, namely, the dynamical modes are correlated in time. EOFs are uncorrelated, but EOFs are not dynamical decay modes. Recall that this would have been the case for the uniform planet where the TDMs are orthogonal and coincide with the EOFs (this holds as well for the case of the infinitely deep mixed layer).

Another interesting property is that the functions are mutually orthogonal.

## Notes for Further Reading

There are many books on stochastic processes. For an introduction, the nicely written and inexpensive book by Bulmer (2002) covers the principles of statistics including calculus. Some math techniques for statistics are in the classic by Cramér (19th printing in 1999). The book by Cramér and Leadbetter (1995) serves as a good introduction to stochastic processes including vector processes. A useful handbook to consult is that of Gardiner (1985). Electrical engineering books often have good coverage of stochastic methods, e.g., Gardner (1989); and the more comprehensive, Papoulis (1984).

## Exercises

**9.1**Let us consider a time-dependent energy balance model forced by a sinusoidal forcing:**9.52**where , and are all constants.

**a.**Let the solution (temperature) of the energy balance model be in the form**9.53**- Set up the energy balance model for the temperature field given above and derive the temperature field.
**b.**How do the amplitude and the phase of the temperature field depend on the frequency of forcing?

**9.2**Let us consider a time-dependent energy balance model forced by noise:**9.54**where , and are all constants.

**a.**Let the solution (temperature) of the energy balance model be in the form**9.55**- Set up the energy balance model for the temperature field given above and derive the temperature field.
**b.**Find the phase lag of the temperature field with respect to the noise forcing.

**9.3**Find the equilibrium solution of the energy balance model forced by an impulsive radiative forcing:**9.56**where the impulsive radiative forcing is given by

Note that , and are all constants.

**9.4**Let us consider a time-dependent energy balance model forced by an impulsive radiative forcing:**a.**Show that the Fourier transform of the impulsive radiative forcing is given by**9.58****9.59****b.**Given the result in Part (a), determine the solution in the form**9.60****c.**In the limit of , show that the solution is of the form**9.61****d.**Determine the lag of the solution as a function of frequency . Then, show that the time-dependent solution approaches the equilibrium solution in Exercise 9.3.

**9.5****a.**Let be spatially white noise forcing. Show that its expansion coefficients satisfy**9.62****b.**If the noise forcing is white both spatially and temporally, show that the expansion coefficients of satisfy**9.63**- which is the desired relationship.

**9.6**Let us consider a time-dependent energy balance model for anomalous temperature due to a noise forcing:**9.64**where is white both in space and time (see Exercise 9.5).

**a.**Using spherical harmonics and Fourier basis functions, obtain the solution of the energy balance model forced by spatially and temporally white noise forcing.**b.**Determine the spectral density function of the temperature response.**c.**Show that**9.65**- On the basis of the expression above, derive an expression for spatial covariance and contemporaneous spatial variance at a point , which is essentially an integration of the spectrum with respect to and .
**d.**Show that**9.66**- where
**9.67**

**9.7**Let us consider the solution of an energy balance model forced by an arbitrary but steady forcing:**9.68****a.**Using the Green's function method, show that the solution of the energy balance model above is given by**9.69**- where the Green's function is the solution of the equation
**9.70** **b.**Obtain the Green's function in Part (a) in terms of spherical harmonics.**c.**Using the Green's function obtained in Part (b), determine the solution of the given energy balance model.

**9.8**Let us consider an energy balance model forced by :**9.71**Note that the parameters and are now functions of position. Recast the given energy balance model in a spectral form using spherical harmonics and Fourier functions as basis sets in space and time, respectively.

**9.9**Consider an energy balance model forced by :**9.72**where the parameters and are now functions of position. Assume that

**9.73****a.**Rewrite the homogeneous form of the energy balance equation above in terms of by using the assumed form of the solution. The resulting equation should be in the form of an eigenvalue problem. Discuss the orthogonality properties of the resulting eigenfunctions.**b.**Determine the solution of the given energy balance model in terms of the eigenfunctions derived in Part (a).