Archana C.Varsoliwala,Twinkle R.Singh

Applied Mathematics and Humanities Department,Sardar Vallabhbhai National Institute of Technology,Surat,395 007,Gujarat,India

Keywords:Atmospheric internal waves Shallow-fluid equations System of nonlinear PDEs Elzaki Adomian decomposition method Climate prediction

ABSTRACT This article involves the study of atmospheric internal waves phenomenon,also referred to as gravity waves.This phenomenon occurs inside the fluid,not on the surface.The model is based on a shallow fluid hypothesis represented by a system of nonlinear partial differential equations.The basic assumption of the shallow flow model is that the horizontal size is much larger than the vertical size.Atmospheric internal waves can be perfectly represented by this model as the waves are spread over a large horizontal area.Here we used the Elzaki Adomian Decomposition Method (EADM) to obtain the solution for the considered model along with its convergence analysis.The Adomian decomposition method together with the Elzaki transform gives the solution in a convergent series without any linearization or perturbation.Comparisons are built between the results obtained by EADM and HAM to examine the accuracy of the proposed method.

1.Introduction

Internal waves also known as gravity waves,is a phenomenon which occurs inside a geophysical fluid(like atmosphere,oceans or lakes),rather than on the surface of it.For the phenomenon to occur,the fluid must be layered-each layer must have the same density and temperature and this should vary only with height.If the density varies over a small vertical height,the waves propagate horizontally like surface waves [14].Atmospheric internal waves form when a uniform layer of air moves over a large barrier like a mountain range.When the air hits the barrier,horizontal stripes of homogeneous air are disturbed,which forms a wave pattern.They can lead to the creation of wave clouds-clouds formed as steady air flows over a barrier such as a mountain [4,33].As the mass of air propagate through the wave,it goes through repeated ascent and decline.If there is adequate humidity in the atmosphere,clouds will form at the cooled peaks of these waves.In the descendant part of the wave,those clouds will vaporize due to adiabatic heating,causing the typical clouded and clear stripes.This phenomenon is at times noticed in different parts of the world,as shown in Fig.1.Internal waves phenomenon in Australia,causes the formation of Morning Glory Clouds,in the southern part of the Gulf of Carpentaria.Fig.2 shows satellite image of Morning Glory Clouds over the Gulf of Carpentaria.Recent studies indicate that these waves have a significant impact on atmospheric temperature,chemistry and turbulence,and are progressively employed in atmospheric simulations.Climate prediction is part of numerical weather prediction which aims to provide a generic prediction of climate over long periods like couple of weeks to few years,which would help in planning agriculture,minimize the effects of drought,cyclone forecasting and air quality forecast [34].Current climate models have limited ability to forecast which includes a period of few days only.It is difficult to realistically represent the complex interactions that happen among the ocean,atmosphere and land surface in such models.Climate models are generally models of the atmosphere and ocean,which are systems of partial differential equations,based on laws of physics,which is a combination of fluid motion,thermodynamics,and chemistry.Since the atmosphere is a fluid,it is important to study its properties for better climate prediction.Studying the functioning of these waves would enable more accurate simulations for weather prediction.

Shallow-fluid equations,also known as shallow water equations,are the usual model to represent the phenomenon of atmospheric internal waves.The main presumption of the shallow-fluid model is that the horizontal dimension is much greater than the vertical dimension.Shallow water equations are systems of nonlinear partial differential equations used to describe fluid flows in oceans,coastal areas,and the atmosphere.Rossby waves [7]and Kelvin [6]waves in the atmosphere are modeled by shallow water equations.Fig.3 shows the schematic representation of internal waves.

Earlier some researchers have discussed the internal waves phenomenon from different viewpoints.Chakraverty and Karunakar[17]have obtained the solutions of linear and non-linear shallow water wave equations using Homotopy Perturbation Method(HPM).Imani et al.[14]have used Reconstruction of Variational Iteration Method (RVIM) for computing the coupled Whitham–Broer–Kaup shallow water.Busrah et al.[4]applied Homotopy and Variational Iteration Methods to solve the atmospheric Internal Waves.Jaharuddin and Hermansyah [13]investigated internal waves of the ocean involving the tidal force.Tort and Dubos[23]presented dynamically consistent shallow-atmosphere equations with a complete Coriolis force.Goswami et al.[12]analyzed analytical techniques for fractional partial differential equations occurring in ion acoustic waves in plasma.Stewart and Dellar[22]highlighted hyperbolicity of multilayer shallow water equations with complete Coriolis force.Le Roux et al.[19]have applied Finite Elements for shallow water equations ocean models.Boyd and Zhou [5]observed effects of Kelvin waves in the nonlinear shallow water equations on the sphere.Kumar [18]derived an analytical solution of the coupled one-dimensional time fractional nonlinear shallow water equation.Also,several authors have applied different techniques to solve differential equations from various angles and perspective [28–32].

Elzaki Adomian Decomposition Method(EADM) is mixture of two methods-Elzaki Transform and Adomian Decomposition Method.Tarig Elzaki was established Elzaki transform in 2011 and George Adomian was introduced Adomian Decomposition Method(ADM) in 1981 [1,2].Elzaki Adomian Decomposition Method is used to solve non-linear PDE without discretizing them,with less computation which leads to more realistic representation.Already many researchers have applied EADM to find the solution of different kinds of problems [3,15,16,21,24,25,27].

The primary purpose of this section is to expand the implementation of Elzaki Adomian Decomposition Method (EADM),to analyze the behavior of shallow fluid equations which is a system of non-linear partial differential equations,and its applications to the phenomenon of atmospheric internal waves that propagate within a fluid medium.

2.Model formulation

Atmospheric internal waves are modeled by a system of nonlinear partial differential equations,based on shallow fluid assumption.The basic equations of fluid motion in differential form are obtained by conservation of mass and momentum.Shallow fluid refers to the fact that the depth of the fluid layer is small as compared to its height.Here,the atmosphere is assumed to be a homogeneous fluid-meaning its density does not vary in space,autobarotropic-meaning its density is a function of pressure only,hydrostatic,incompressible and inviscid.The basic momentum equations are as follows [26]

Fig.1.Wave cloud pattern formed over the Nouvelle Amsterdam.

Fig.2.Morning Glory Clouds over the Gulf of Carpentaria.

The continuity equation is

For incompressible and homogeneous fluid,

So thatρ=ρ0,whereρ0is a constant.Therefore,Eq.(4) becomes

The hydrostatic equation can be derived as

Differentiating Eq.(7) with respect tox,

which implies that there is no horizontal fluctuation of the vertical pressure gradient or vertical fluctuation of the horizontal pressure gradient-by virtue of barotropy.As the wind produced by the pressure-gradient force and the consequent Coriolis force,all forces are invariable with depth.Integrating Eq.(7) over the depth of the fluid,

PTandPBrepresent the pressure at the top and bottom edge of the fluid respectively.

Hereηis the depth of the fluid.IfPT=0,orPT?PB,

A new representation of the pressure-gradient term in Eqs.(1) and(2) can be obtained under the proposition that the horizontal pressure gradient at the base of the fluid is proportional to the gradient in the depth of the fluid.Integrating the incompressible continuity equation (Eq.(6)) with respect toz,the equation becomes:

Since the pressure gradient is not a function ofz,and assumingmandpare not a function ofzinitially,their derivatives would neither be a function ofz.This gives

forz=η.The vertical velocity at the base of the fluid is 0.Also,

Therefore,the shallow-water continuity equation becomes

We now have 3 equations in terms ofm,pandη.

A one-dimensional form of the equations in terms ofm,pandηare considered.Specifying a constant pressure gradient of desired magnitude in theydirection,allows us a meanmcomponent on which perturbations occur.The system of equation becomes

Fig.3.Schematic representation of internal waves.

wherexis a space coordinate,tis time,the independent variablesmandpare the cartesian velocities,ηis depth of a fluid,frepresents the Coriolis parameter,gis acceleration of gravity,ˉHrepresents mean depth of a fluid,and ˉUis the specified,constant mean geostrophic speed [4,26]().

3.Elzaki transform

Tarig Elzaki was introduced the Elzaki transform and it is based on Fourier integral.This transform is used to obtain the solution of differential equations in the time domain [8,9].

Consider the setA,where

Mmust be constant finite number andω1,ω2 may be finite or infinite in above setAandu(t)is given byE[u(t)]=U(p)and defined as

4.Description of Elzaki Adomian decomposition method(EADM)

The general nonlinear partial differential equation [27]is

The 2nd order linear operator is given byL=?2/?τ2,the linear operator of order less thanLis given byR,the nonlinear operator isNand source term is given byg(r,τ).

Apply Elzaki transform on Eq.(21)

Eis Elzaki transform.Employing the properties of Elzaki transform in (23)

Apply inverse Elzaki transform to Eq.(24)

The term appearing from the source and the specified initial conditions is represented byG(r,τ).

The series solution of Eq.(25) is

The nonlinear operator is given as

whereAnare the Adomian polynomials and formula (28) is used to obtain polynomials.

substituting Eqs.(26) and (27) into Eq.(25),

Comparing on both sides of the Eq.(29)

In general,

The solution is

5.Convergence analysis

Theorem5.1.LetN:H→Hbeanonlinearoperator.HdenotesHilbertSpace[20]andletwbeanexactsolutionofEq.(21).whichisobtainedbyEq.(32),convergestow,if?γ,0≤γ＜1,such that‖wk+1‖≤γ‖wk‖,?k∈N∪ {0}.

Proof.We have

‖F(xiàn)n+1-Fn‖=‖wn+1‖≤γ‖wn‖≤γ2‖wn-1‖≤...≤γn+1‖w0‖.

However for ?n,m∈N,n≥m,we have

Definition5.2.For ?i∈N∪ {0},γican be obtained as

Corollary5.1.convergestoexactsolutionw,when0≤γi＜1,i=1,2,3...

6.Application of Elzaki Adomian decomposition method

The suitable initial condition of above problem is taken as [4]

Performing the Elzaki transform on Eq.(18),

On using the initial condition and then employing the inverse Elzaki transform,

Applying Adomian decomposition approach,

Comparing the results on both sides of Eqs.(39)–(41),

whereAn,Bn,Cn,DnandFnare the Adomian polynomials given bymmx,ηx,mpx,mηxandηmxrespectively.

Some Adomian polynomials are given by

Employing Adomian polynomials (45) and the iteration formulas(42),(43) and (44),the approximate solutions upto three terms are given by Eqs.(46)–(48).The following parameters are used for the

numerical solutions [4]: Coriolis parameterf=2Ωsinα,whereΩ=7.29×10-5rad/sandα=constant of gravityg=9.8m/s2and constant of pressure gradient of desired magnitudewhere=2.5m/sis specified.

Fig.4.Solution of m ( x,t).

Fig.5.Solution of p ( x,t).

Fig.6.Solution of η( x,t).

Fig.7.Numerical solutions of m,p,η for t=1 and 0≤x≤2.

Fig.8.Numerical solutions of m,p,η for x=1 and 0≤t≤0.1.

Applying Corollary 5.1 for convergence analysis,

7.Results and discussion

Here MATHEMATICA is used to obtain the numerical results.Table 1 shows the numerical solutions for horizontal velocity componentm,vertical velocity componentpand depthηfor different time valuest=0,0.02,0.04,0.06,0.08,0.1 and space valuesx=0.2,0.4,0.6,0.8,1,1.2,1.4,1.6,1.8,2.Further,from Table 1,it can be observed that values ofηare very small as compared tomandp,which is fit with atmospheric internal waves phenomenon,

Table 1 Numerical solutions of m,p and η for different times and spaces.

Table 2 Numerical comparison of solutions of m.

Table 3 Numerical comparison of solutions of p.

Table 4 Numerical comparison of solutions of η.

Fig.9.Approximate solution of m ( x,t).

Fig.10.Approximate solution of p ( x,t).

Fig.11.Approximate solution of η( x,t).

where the fluid’s depth is negligible in comparison to its horizontal scale.Additionally,Tables 2,3,and 4 describe the comparison of the results of EADM and HAM.It also indicates a very good agreement between the results derived by EADM and HAM.EADM requires less number of parameters,no discretization and linearization as compare to other analytical methods.Whereas HAM has limitations,including the introduction of auxiliary operators,auxiliary parameters,and auxiliary functions.So far,there is no rigorous mathematical theory to guide the selection of auxiliary operators,auxiliary parameters,or auxiliary functions.Thus EADM is better effective in approximating the proposed phenomenon than HAM.Also,graphical comparison of these two methods are shown in Figs.4 5 and 6 respectively which shows the accuracy of the proposed method.Figures 7 and 8 depict the numerical solutions ofm,pandηfor 0≤x≤2,t=0.1 and 0≤t≤0.1,x=1 respectively.Figures 9,10 and 11 show the 3D representation ofm,p,andη.

8.Conclusion

In this work,we have successfully applied EADM method to find an approximate solution of atmospheric internal waves phenomenon along with its convergence analysis.The convergence analysis shows that the series solutions obtained are convergent hence,the solutions are well defined.Solutions derived by EADM are compared with HAM.Graphical comparison of these two methods are shown in Fig.4,5 and 6 respectively.It shows that all solutions are in perfect agreement.EADM provides solutions which are efficient computationally,as there is no need to linearize the equations.Hence it tends to provide a realistic portrayal of the said model.Figures 7 and 8 depict the numerical solutions ofm,pandηfor 0≤x≤2,t=0.1 and 0≤t≤0.1,x=1 respectively.Figures 9,10 and 11 show the 3Drepresentation ofm,p,andη.From the results,we can conclude that EADM is effective in approximating the phenomenon of atmospheric internal waves.The obtained results provide a thorough and pragmatic look into the behaviour of these waves,which can be leveraged to refine weather and climate models,as the waves have a significant influence on climate change,as has been observed in recent years.

Declaration of Competing Interest

All authors have participated in (a) conception and design,or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.

This manuscript has not been submitted to,nor is under review at,another journal or other publishing venue.

The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript

Acknowledgements

The authors are very much thankful to S.V.National Institute of Technology,Surat for providing great opportunity for research work.

Appendix A.Some of the properties of Elzaki transform[8–11]

Appendix B.The algorithm of Elzaki Adomian decomposition method

Here we present an algorithm that compute the solution of problem (21).

Algorithm

Consider the problem (21).

Step 1.Apply Elzaki transform (E) on both sides of Eq.(21).

Step 2.Apply the property of the Elzaki transform.

Step 3.Substitute the given initial conditions (22) in the equation.

Step 4.Apply the inverse Elzaki transform (E-1) to the equation.

Step 5.Get first term asG(X,T)which represents the term arising from the supply term and the prescribed initial conditions.

Step 6.Write the solution of problem asand nonlinear operator is defined as

Step 7.Calculate the Adomian polynomials for the nonlinear term by formulaAn=0,1,2,...

Step 8.After equating the results on both sides of the equation,calculateUn+1(X,T)in a loop.

Step 9.Construct the solution using the seriesU(X,T)=