research article the determination of feasible control...

Research ArticleThe Determination of Feasible Control Variables forGeoengineering and Weather Modification Based on theTheory of Sensitivity in Dynamical Systems

Sergei A Soldatenko and Rafael M Yusupov

St Petersburg Institute for Informatics and Automation The Russian Academy of Sciences No 39 14th LineSt Petersburg 199178 Russia

Correspondence should be addressed to Sergei A Soldatenko soldatenkoiiasspbsu

Received 25 November 2015 Revised 2 May 2016 Accepted 23 May 2016

Academic Editor Ai-Guo Wu

Copyright copy 2016 S A Soldatenko and R M Yusupov This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Geophysical cybernetics allows for exploring weather and climate modification (geoengineering) as an optimal control problemin which the Earthrsquos climate system is considered as a control system and the role of controller is given to human operators Inmathematicalmodels used in climate studies control actions thatmanipulate theweather and climate can be expressed via variationsinmodel parameters that act as controls In this paper we propose the ldquoinstability-sensitivityrdquo approach that allows for determiningfeasible control variables in geoengineering The method is based on the sensitivity analysis of mathematical models that describevarious types of natural instability phenomenaThe applicability of this technique is illustrated by amodel of atmospheric baroclinicinstability since this physical mechanism plays a significant role in the general circulation of the atmosphere and consequently inclimate formationThe growth rate of baroclinic unstable waves is taken as an indicator of control manipulations The informationobtained via calculated sensitivity coefficients is very beneficial for assessing the physical feasibility of methods of control of thelarge-scale atmospheric dynamics and for designing optimal control systems for climatic processes It also provides insight intopotential future changes in baroclinic waves as a result of a changing climate

1 Introduction

Weather modification is the operation of deliberately alteringthe atmosphere that leads to changes in the natural evolutionof physical and dynamical atmospheric processes Predom-inantly weather modification is successfully accomplishedvia cloud seeding in order to affect precipitation for thepurpose of the local water supply (eg [1ndash4])Many countriescurrently practice cloud seeding operationally The nextsuccessful example of weather modification is fog dispersalto improve visibility at airports by heating or seeding [5]Some attempts taken in different countries to reduce damagefrom hazardous weather events such as hurricanes strongtornado and thunderstorm winds hail lighting and floodsunfortunately have not been so successful The detailedretrospective review and the current status of weathermodification with the findings and relevant concepts wereconsidered for example in [6ndash8] Over the last decades

the weather modification has transitioned from local scaleoperations to a global weather modification known also asgeoengineering Geoengineering or climate engineering isa deliberate and purposeful large-scale modification of theEarthrsquos climate system (ECS) and first of all the atmospherewhich is the most unstable and fast-changing element of theECS [9] Geoengineering was offered by scientific commu-nity as a response to global warming which is happeningSince mankind is causing global warming by anthropogenicCO2emissions [10] the most obvious idea to reduce the

consequences of global climate change is a sequestration ofanthropogenic greenhouse gas (GHG) emissions Howeverthis is unlikely achievable in foreseeable future due to the con-tinuing growth of the world economy and population Scien-tists and engineers proposed several solutions to stabilize theglobal climate (eg [11 12]) These solutions can be dividedinto two main categories carbon dioxide removal technolo-gies (CDR) and solar radiation management (SRM) CDR

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2016 Article ID 1547462 9 pageshttpdxdoiorg10115520161547462

2 Journal of Control Science and Engineering

technologies include for example various engineered solu-tions that remove GHGs directly from the atmosphere usingbig machines or chemical absorbents SRM techniques aimmainly to reflect some percentage of the solar radiation backinto the outer space This can be achieved by changing theplanetary albedo (increasing the reflectiveness of the Earthrsquossurface or clouds) dispersing small particles into the strato-sphere or deploying mirrors in the upper atmosphere toreflect sunlight However all of these technologies introduceuncertainties and unexpected consequences that must beexplored

Let both weather modification and climate engineeringhereafter be referred to as the geoengineering Realization ofgeoengineering projects is a purposeful process that is theprocess imminently connected with a specific objective thatcan be formulated in various ways In this context geoengi-neering is per se the process of controlling the ECS Howevergeoengineering is still considered outside of the scope ofcontrol theory as an intentional action to influence naturalprocessesMeanwhile in the late 1970s a uniformmethodologyfor control geophysical processes including processes occur-ring in the ECS was formulated on the basis of the ideasfrom cybernetics by one of the authors of this paper [13] Inthis monograph the concept of geophysical cybernetics wasintroduced as the new research areawithin the control theoryGeophysical cybernetics explores a self-regulating feedbackcybernetic system in which the ECS is considered as thecontrol object and the role of the controller is given to humanoperators From the standpoint of geophysical cyberneticsclimate and weather manipulation represents an optimalcontrol problem which aims to synthesize the control lawthat ensures the achievement of the desired results that canbe expressed in terms of extremal problem [13 14] In ourprevious publications [14ndash16] an optimal control problemfor the ECS has been conceptually formulated in bothprobabilistic and deterministicmanner Let us emphasize thatthe ECS is a unique natural physical system with a largenumber of specific attributes [17] which makes the controlproblem for this system extremely complex The develop-ment of physically feasible methods to control the ECSrequires the determination of feasible control mechanismsand variables The theory of sensitivity in dynamical systems[18 19] serves as a theoretical instrument for solving thisproblem

This paper presents a short description of the ECS as acontrol system emphasizing its unique physical propertiesrelevant to the control problems of large dynamical systemsIt is also noted that the amount of energy released duringprocesses that drive the ECS is orders of magnitudes greaterthan human capabilities which makes the control of the ECSvia direct interventions very problematic However somephysical and dynamical processes occurring in the ECS areinherently unstable which apparently allows one to imple-ment geoengineering projects using significantly less energyresources in comparisonwith the energy of natural processesThus the study of various types of instabilities in ECS is veryimportant for developing an appropriate control strategyTheinstability of natural physical processes is mathematicallystudied as a problem of finding the necessary conditions for

the growth of the infinitesimal perturbations Since theseconditions are expressed in terms of some model parame-ters by using sensitivity analysis we can first determineparameters that can be considered as controls and secondmake a conclusion on the hypothetical possibility of controlof physical process under consideration To illustrate thisldquoinstability-sensitivityrdquo approach we as an example con-sider the atmospheric baroclinic instability as the controlledobject since this type of hydrodynamic instability plays asubstantial role in the formation of large-scale atmosphericeddies in the extratropical atmosphere and therefore in thegeneral circulations of the atmosphere and climate [20] Toexplore the response of baroclinic instability to geoengi-neering interventions a multilayer geostrophic model of theatmosphere is appliedThe growth rate of baroclinic unstablewaves is taken as an indicator of control manipulations Theinformation obtained via calculated sensitivity coefficientsis very beneficial for assessing the physical feasibility ofmethods of control of the large-scale atmospheric dynam-ics and for designing optimal control systems for climaticprocesses

2 Climate System as a Unique Control System

TheECS is a unique and peculiar natural physical system thatis extremely difficult to control since it possesses a number ofspecific properties including but not limited to the following[14 17 21 22]

(i) The ECS is a complex interactive system with a widevariety of positive and negative feedback mechanisms TheECS consists of the atmosphere ocean sea-ice land surfaceand other bodies of water and includes global carbon cyclechemistry and aerosols These natural subsystems have sub-stantial differences in their physical and chemical propertiesstructure and behavior they can be strongly or weaklycoupled and linked together by means of coupling physicalmechanisms

(ii) Dynamical and physical processes in the ECS occurover a broad spectrum of scales in both space and timeTime scales are varied from seconds (turbulent fluctuations)to dozens of years (climate change and variability) Since theECS is a global system its spatial spectrum of motions coversmolecular to planetary scales It is important that dynamicalprocesses in the atmosphere and ocean are nonlinear andchaotic

(iii) Processes in the ECS oscillate due to bothinternal factors (natural oscillations) and external forcing(forced oscillations) Natural oscillations are due to theinternal instability of ECS with respect to stochasticinfinitesimal disturbances Human impact on the ECS bothintentional and unintentional is considered as externalforcing

The ECS has certainly a number of other specificattributes that make it a unique and complex physical systemOne of the most effective instruments in studying the ECSis mathematicalnumerical modeling Climate mathematicalmodels used in a variety of applications are commonlydeterministic and derived from a set of multidimensional

Journal of Control Science and Engineering 3

nonlinear differential equations in partial derivatives whichare the equations of fluid dynamics and thermodynamicsModels also take into consideration the specific properties ofECS as well as its cycles such as water nitrogen oxygen andcarbon cycles

An optimal control problem for the ECS remains poorlystudied due to its relative novelty and enormous complexity[15 16] To develop a general framework for optimal controlof the ECS the following should be taken into consideration

(i) The ECS is a spatially distributed system thereforethe control actions for manipulating this system should alsobe distributed in space However implementation of suchcontrols is weakly developed

(ii) Processes in the ECS possess enormous energypotential It is hardly possible to provide control actionswhose energy is comparable to the energy of natural physicalprocesses Therefore the identification of sensitive pointsin which the ECS is in an unstable state is a criticalissue

(iii) Large-scale and huge energy of climate-driven pro-cesses impose very strict requirements for the accuracy andreliability of control systems since even minor errors incontrol actions can be disastrous

(iv) Processes in the ECS are interconnected thereforechanges in the dynamics of some processes can result inuncontrollable consequences

(v) Control actions to perform geoengineering andweather modification operations must be physically feasibleand executable

Consequently discussing geoengineering within thescope of optimal control theory we are faced with anumber of problems including the problem of choosingvariables that can be considered as controls Mathematicalclimate models incorporate a certain number of physi-cal processes responsible for the transformation of energyThese processes represent natural control mechanisms ofthe ECS that can be considered as potential artificialcontrol mechanisms Unfortunately these physical mech-anisms cannot be explicitly identified and represented inclimate models due to modelsrsquo discrete spatial-temporalstructure and can only be described parametrically undersome simplified assumptions Some of the newly introducedparameters together with parameters that describe the exter-nal forcing can be considered as control variables (param-eters) Then geoengineering actions can mathematically berepresented via variations in parameters that act as controlsBy using sensitivity analysis we can explore the influenceof variations in control parameters on the behavior ofthe ECS and therefore evaluate the hypothetical possi-bility of various physical mechanisms to control the ECSHowever we need to keep in mind that control meth-ods should be on the one hand physically feasible andon the other hand technically executable Many physicalprocesses occurring in the ECS are inherently unstable[20 23] This gives the possibility to control the ECS vianatural instability mechanisms Apparently the atmospherewhich is the most rapidly moving and changing componentsof the ECS represents the most suitable system to becontrolled

3 The Model of Baroclinic Instabilityin the Atmosphere

Let us consider the set of the so-called primitive equationsin isobaric coordinate system commonly used in modellinglarge-scale atmospheric flows [23]

120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910+ 120596

120597119906

120597119901minus 119891V = minus

120597Φ

120597119909 (1)

120597V120597119905

+ 119906120597V120597119909

+ V120597V120597119910

+ 120596120597V120597119901

+ 119891119906 = minus120597Φ

120597119910 (2)

119877119879

119901= minus

120597Φ

120597119901 (3)

120597119906

120597119909+

120597V120597119910

+120597120596

120597119901= 0 (4)

120597119879

120597119905+ 119906

120597119879

120597119909+ V

120597119879

120597119910minus

119877119879

119892119901(120574119889minus 120574) 120596 = 0 (5)

Here 119906 and V are the horizontal velocity components in thedirections 119909 and 119910 respectively (the horizontal coordinatesx and y are directed eastward and northward) 120596 equiv 119889119901119889119905 ispressure vertical velocity where 119901 is pressureΦ is the geopo-tential T is the temperature R is the gas constant for dry airf is the Coriolis parameter 119892 is the gravity acceleration 120574

119889

is the dry adiabatic lapse rate 120574 is the reference state lapserate We will employ the 120573-plane approximation so that theCoriolis parameter 119891 is represented as 119891 = 119891

0+ 120573119910 where

1198910is a standard value of the Coriolis parameter at the mid-

latitude and 120573 = 120597119891120597119910 is the latitudinal gradient of 119891Let us make some comments regarding the set of

equations (1)ndash(5) These are nonlinear differential equationsthat are used to describe adiabatic large-scale atmosphericdynamics Equations (1) and (2) are the momentum equa-tions which mathematically express Newtonrsquos second law ofmotion Equation (3) is a hydrostatic equationThe continuityequation (4) expresses the conservation of mass and (5)is a thermal energy equation representing the first law ofthermodynamics

Baroclinic instability is commonly explored by linearizingthe model equations around some unperturbed reference(basic) flow and then solution of the problem can be foundusing initial-value or eigenvalue approachesWe suppose thatthe atmospheric reference flow defined by 119906 V 120596 119879 Φis geostrophic (ie the Coriolis force and pressure gradientforces are in balance) and satisfies the following relations

119906 = minus1

1198910

120597Φ

120597119910

V = 0

120596 = 0

120597Φ

120597119901= minus

119877119879

119901

(6)


Pressure (hPa)0

250

500

750

1000

0

1

2

3

4

Level1205960

1205962

1205964

1205951

1205953

Figure 1 The distribution of levels and arrangement of variables inthe vertical for two-layer model

where 119879 = 119879(119910 119901) The reference state (6) is a solution of(1)ndash(5) that describes the zonal flow

120597119906

120597119901=

119877

1198910119901

120597119879

120597119910 (7)

which matches the specified distribution of the zonallyaveraged temperature 119879(119910 119901) and represents thermal windbalance Let us underline that the geostrophic approximationused in this study is valid to high accuracy for the large-scaleatmospheric flows Then with geostrophic assumption (1)ndash(5) reduced to the vorticity equation and the thermodynamicequation [23]

(120597

120597119905+ 119906

120597

120597119909+ V

120597

120597119910)nabla2120595 + 120573

120597120595

120597119901= 1198910

120597120596

120597119901 (8)

(120597

120597119905+ 119906

120597

120597119909+ V

120597

120597119910)

120597120595

120597119901+

120590

1198910

120596 = 0 (9)

Here 120595 = Φ1198910is a geostrophic stream function and 120590 =

(11987721198791198921199012)(120574119889minus 120574) is the static stability parameter where 119892 is

the gravity acceleration To conserve the mass the followingboundary conditions are used for the pressure velocity [23]

120596 = 0 at 119901 = 0 119901 = 1198750 (10)

where 1198750is a standard pressure at the Earthrsquos surface

In this paper wewill consider an analytical solution of thebaroclinic instability problem using an eigenvalue approachfor a two-layer model The model vertical structure is shownin Figure 1 For this model the boundary condition (10) gives1205962= 1205964= 0 Applying quasi-geostrophic vorticity equation

(8) to the 750 and 250-hPa surfaces and approximating thederivative 120597120596120597119901 by finite differences one can obtain theresulting vorticity equations at levels 1 and 3

120597

120597119905nabla21205951+ u1sdot nabla (nabla

21205951) + 120573

120597120595

120597119909=

1198910

Δ1199011205962 (11)

120597


21205953) + 120573

120597120595

120597119909= minus

1198910

Δ1199011205962 (12)

where u = 119906119894+V is the horizontal velocity andΔ119901 = 500 hPaThe thermodynamic energy equation (10) is applied at level 2

120597

120597119905(1205951minus 1205953) + u2sdot nabla (120595

1minus 1205953) minus

120590Δ119901

1198910

1205962= 0 (13)

Thus we have a system of three equations (11)-(13) in thethree variables 120596

2 1205951 and 120595

3 To study the instability of the

basic zonal flow with respect to infinitesimal perturbationsthese equations are linearized around the basic state (6) Letus assume that

1205951= minus1199061119910 + 120595

1015840

1(119909 119905)

1205953= minus1199063119910 + 120595

1015840

3(119909 119905)

1205962= 1205961015840

2(119909 119905)

(14)

Substituting (14) into (11)ndash(13) defining

119906119898

=1

2(1199061+ 1199063)

119906119879=

1

2(1199061minus 1199063)

120595119898

=1

2(1205951015840

1+ 1205951015840

3)

120595119879=

1

2(1205951015840

1minus 1205951015840

3)

(15)

and eliminating the variable 12059610158402yield the following perturba-

tion equations

(120597

120597119905+ 119906119898

120597

120597119909)

1205972120595119898

1205971199092+ 120573

120597120595119898

120597119909+ 119906119879

120597

120597119909(1205972120595119879

1205971199092) = 0

(120597

120597119905+ 119906119898

120597

120597119909)(

1205972120595119879

1205971199092minus 21205822120595119879) + 120573

120597120595119879

120597119909

+ 119906119879

120597

120597119909(1205972120595119879

1205971199092+ 21205822120595119879) = 0

(16)

where 1205822

= 1198912

0[120590(Δ119901)

2] We will seek normal mode

solutions of the following form

120595119898(119909 119905) = Ψ

119898119890119894119896(119909minus119888119905)

120595119879(119909 119905) = Ψ

119879119890119894119896(119909minus119888119905)

(17)

where Ψ119898

and Ψ119879are the amplitude of perturbations k

is a wavenumber and 119888 is a complex phase velocity Bysubstituting (17) into (16) after some algebraic manipulationwe can obtain the following equation for the phase speed ofbaroclinic waves

119888 = 119906119898minus

120573

1198962

1198962+ 1205822

1198962 + 21205822plusmn radic120575 (18)

where

120575 =12057321205824

1198964 (1198962 + 21205822)2+ 1199062

119879

1198962minus 21205822

1198962 + 21205822 (19)

Perturbations will grow exponentially if 119888 has an imaginarypart 119888119894 This will occur if 120575 lt 0


120001000080006000400020000

Wavelength (km)

0

5

10

15

20

25

30

35

40

Ther

mal

win

d (m

middotsminus1)

ci gt 0

ci = 0

Figure 2 Instability diagram displaying wavelength regions ofstable and unstable waves

Then the increment of growingmode with a wavenumber119896 is given by the following expression

120594119896equiv 119896119888119894=

radic100381610038161003816100381612057321205824 + 1199062

1198791198964 (1198964 minus 41205824)

1003816100381610038161003816

119896 (1198962 + 21205822) (20)

This equation shows that the growth rate of unstable pertur-bations depends on the wind shear 119906

119879associated with the

pole-equator temperature gradient and the variable 120582 whichis a function of the static stability 120590 Thus parameters 119906

119879and

120590 can be considered as feasible controls

4 Sensitivity Analysis of BaroclinicInstability in the Context of AtmosphericDynamics Control

First let us highlight the most important properties of themodel described in Section 3 By setting the discriminant (19)equal to zero the so-called marginal stability curve (neutralcurve) that separates the stable region from the unstableregion can be plotted as function of the mean thermal wind119906119879 and perturbation wavenumber 119896 (see Figure 2)

1199062

119879=

12057321205824

1198964 (41205824 minus 1198964) (21)

Figure 2 shows that there are two stable regimes one for shortwaves and another one for long wavesThus two-layer modelhas a shortwave cut-off (119871SC) and longwave cut-off (119871LC)any particular wave is unstable if its length 119871

119909satisfies the

following double inequalities 119871SC lt 119871119909lt 119871LC The range of

unstable waves depends on static stability and thermal windand can be found from (21) by the following way

1198964= 21205824plusmn (4120582

8minus

12057321205824

1199062119879

)

12

(22)

If the thermal wind is less than the minimum value 119906min119879

on the stability curve then all waves are stableTheminimum

thermal wind required for the development of instability canbe estimated by differentiating (21) with respect to 119896 |119906min

119879| gt

120573(21205822) If the thermal wind exceeds the value of 119906min

119879 then

both stable and unstable waves can existThe dispersion diagram (Figure 3) shows that baroclinic

waves of different wavelengths travel in space at differentvelocities In Figure 3 two values of the phase velocity corre-spond to stable waves and one value to unstable waves Shortand longwaves both stable and unstable travel eastward Onlyvery long waves (119871

119909gt 6000 km) can propagate westward

however these waves are outside of our interest Phase speedof quasi-barotropic stable Rossby waves is also shown inFigure 3 for comparison These results are obtained for 119891

0=

103 sminus1 that matches the latitude 1205930= 45∘N 120573 = 163 times

10minus11mminus1 sminus1 120590 = 2 times 10

minus6m2 Paminus2 sminus2 119906119879

= 75msminus1 and119906119898

= 15msminus1 [23]In the two-layer model the static stability parameter 120590

and the vertical wind shear 119906119879control the development of

baroclinic instability To estimate the influence of controlactions of the development of baroclinic instability thesensitivity coefficients 119878

120590and 119878

119906119879are employed Analytical

expressions for 119878120590and 119878119906119879are obtained by differentiating (20)

with respect to 120590 and 119906119879

119878120590equiv

120597120594119896

120597120590= minus

1198961205822

120590 (1198962 + 21205822)2

21199062

1198791198964(1198964+ 21205822) minus 12057321205822

11990621198791198964 (1198964 minus 41205824) + 12057321205824

sdot radic11990621198791198964 (1198964 minus 41205824) + 12057321205824

119878119906119879

equiv120597120594119896

120597119906119879

= minus1199061198791198963(1198962minus 21205822)

sdot

radic100381610038161003816100381612057321205824 + 1199062

1198791198964 (1198964 minus 41205824)

1003816100381610038161003816

12057321205824 + 11990621198791198964 (1198964 minus 41205824)

(23)

Sensitivity coefficients 119878120590and 119878

119906119879should be estimated in

the vicinity of some reference values of the static stabilityparameter 120590

lowast and thermal wind 119906lowast

119879 respectively which

depend on the chosen weather and climate conditionsSensitivity coefficients 119878

120590calculated for different basic

values of the static stability parameter 120590lowast provide importantinformation regarding the impact of 120590

lowast on the growthrate of unstable waves As shown in Figure 4 the absolutevalues of sensitivity coefficients 119878

120590exponentially increase

with decreasing wavelength for a specified value of 120590lowast As

an example let us consider two waves (A and B) of differentwavelengths 119871(119860)

119909asymp 3 000 and 119871

(119861)

119909asymp 5 000 km respectively

for the case of 120590lowast = 2 times 10minus6m2sdotPaminus2sdotsminus2 and 119906

lowast

119879= 75msdotsminus1

The sensitivity of wave 119860 with respect to the static stabilityparameter 119878

(119860)

120590asymp minus16 is about 12 times the sensitivity

119878(119861)

120590asymp minus135 of wave 119861 in absolute value Thus short

baroclinic unstable waves possess a high sensitivity to theatmospheric static stability the smaller the wavelength thehigher the sensitivity In contrast long unstable waves aremore sensitive to the vertical wind shear but not to the staticstability (Figure 5) The sensitivity of long unstable wave 119861


1000080006000400020000

Wavelength (km)

minus30

minus20

minus10

0

10

20

30

Wav

e spe

ed (m

sminus1)

Figure 3 Phase velocity as a function of wavelength for baroclinic waves (solid line) and for Rossby waves (dashed line) for the case of120590 = 2 times 10

minus6m2sdotPaminus2sdotsminus2 119906119879= 75msdotsminus1 and 119906


minus2

minus4

minus6

minus8

minus10

minus12

minus14

700060005000400030002000

0

Wavelength (km)

Sens

itivi

tyS120590

120590lowast= 10

120590lowast= 15

120590lowast= 20

120590lowast= 25

120590lowast= 30

Figure 4 Sensitivity coefficients 119878120590versus the wavelength for different reference values of static stability parameter 120590lowast times 10

6m2sdotPaminus2sdotsminus2 forthe case of 119906


with respect to the wind shear 119878(119861)119906119879

asymp 011 is about 2 times thesensitivity 119878

(119860)

119906119879asymp 005 of short unstable wave A

If 120575120590 and 120575119906 represent the control actions such that 120575120590 ≪

120590lowast and 120575119906 ≪ 119906

lowast

119879 the unstable wave growth rate changes 120575120594

119896

induced by 120575120590 and 120575119906 are estimated to a first-order accuracyin the following way

120575120594119896(120575120590) equiv 120594

119896(120590lowast+ 120575120590) minus 120594

119896(120590lowast) asymp 120575120590 times 119878

120590

1003816100381610038161003816120590=120590lowast (24)

120575120594119896(120575119906) equiv 120594

119896(119906lowast

119879+ 120575119906) minus 120594

119896(119906lowast

119879) asymp 120575119906 times 119878

119906119879

10038161003816100381610038161003816119906119879=119906lowast

119879

(25)

Suppose the reference value of the static stability 120590lowast is

equal to 2 times 10minus6m2sdotPaminus2sdotsminus1 [23] which is a typical mid-

latitude tropospheric value of the static stability parameterLet us use (24) to estimate the impact of control 120575120590 on thegrowth rates of unstable perturbations assuming that 120575120590 isminus 5 percentage points of 120590lowast Note that this decrease inthe static stability parameter corresponds to a 02 Ksdotkmminus1

increase in the lapse rate 120574 while the standard troposphericvalue of 120574 is 65 Ksdotkmminus1 [23] The mentioned above lapse ratechange may be achieved in various ways for example bychanging the surface albedo Table 1 illustrates changes in thegrowth rates of unstablewaves caused by control120575120590Themostimportant result is that the short unstable wave of wavelength119871119909

asymp 119871(119860)

119909demonstrates the phenomenal change in the

growth rate 120575120594 of nearly 87 per cent compared to the valuethat corresponds to the unperturbed static stability parameter120590lowast Its growth rate reaches the value of 028 dayminus1 The

influence of 120575120590 on the growth rates of unstable perturbationstends to decrease rapidly with increasing wavelength Sosmall perturbations in the static stability lead to the tangiblechanges to the growth rates of short baroclinic unstable waves(119871119909sim 119871(119860)

119909)

The development of baroclinic instability can be partiallysuppressed if the static stability is increased due to controlactionsThe suppression degree in accordance with Figure 4


2000 4000 6000 8000

Sens

itivi

tySU119879

020

015

010

005

000

Wavelength (km)

ulowastT = 6

ulowastT = 8

ulowastT = 10

ulowastT = 12

ulowastT = 14

Figure 5 Sensitivity coefficients 119878119906119879

versus the wavelength for different reference values of thermal wind 119906lowast

119879for the case of 120590 = 2 times

10minus6m2sdotPaminus2sdotsminus2

SR 120590SR u119905

|SR120590 |SRu119905

Lcrx

Wavelength (km)6000500040003000

6

5

4

3

2

1

0

Figure 6 Relative sensitivity coefficients 119878119877

120590and 119878

119877

119906119879versus the

wavelength 119871119909for the case of 120590 = 2 times 10

minus6m2sdotPaminus2sdotsminus2 and 119906119879

=

75msdotsminus1

is also dependent on the wavelength For example if thecontrol 120575120590 is 5 percentage points of 120590lowast then the amplitudeof the short unstable wave 119860 does not grow

Let us examine now the influence of variations in thethermal wind on changes in the growth rates of baroclinicunstable waves using (25) In calculations the referencevalue of vertical wind share 119906

lowast

119879was set equal to 75msdotsminus1

which corresponds to the vertical gradient of wind velocityequal to 003msdotsminus1sdothPaminus1 [23] According to the thermalwind balance this wind shear is generated by the meridionaltemperature gradient equal to 52∘ K per 1000 km Let thecontrol 120575119906

119879be 5 percentage points of the reference value 119906

lowast

119879

Table 1 Changes in the growth rates of unstable waves for the caseof 120575120590 = minus0 05 times 120590

lowast 120590lowast = 2 times 10minus6m2sdotPaminus2sdotsminus1 and 119906

lowast


119871119909km 5000 4000 3500 3200 3150

120594 dayminus1 037 044 037 021 015119878120590

minus134 minus277 minus485 minus1060 minus1531

120575120594 dayminus1 001 002 004 009 013(120575120594120594) times 100 27 45 108 429 867

Table 2 Changes in the growth rates of unstable waves for the caseof 120575119906119879= 005 times 119906

lowast

119879 120590lowast = 2 times 10

minus6m2sdotPaminus2sdotsminus1 and 119906lowast


119871119909km 5000 4000 3500 3200 3150

119878119906119879

011 008 007 005 005120575120594 dayminus1 004 003 003 003 002(120575120594120594) times 100 108 68 81 95 123

Table 2 shows changes in the growth rates of unstable wavescaused by control 120575119906

119879 From this table it follows that the

relative change in growth rates of both short and long wavesis about 10 percent with respect to that of the unperturbedreference value of the wind shear

The results presented in Tables 1 and 2 can be summarizedas follows changing the vertical stratification of the atmo-sphere is the most appropriate mechanism for controlling thebaroclinic instability

Since control parameters 120575120590 and 120575119906119879

have differentdimensions and different equivalence classes in order tocompare their relative role in changing 120575120594 we can use relative(normalized) sensitivity coefficients

119878119877

120590= 119878120590

120590

120594119896

119878119877

119906119879= 119878119906119879

119906119879

120594119896

(26)


The analysis of the relative sensitivity coefficients leads toan important conclusion There is a critical value of thewavelength 119871

cr119909that divides the spectrum of unstable waves

into two parts (Figure 6) The development of baroclinicinstability ismainly affected by the atmospheric static stabilityfor the case of short-wavelengths (119871

119909lt 119871

cr119909) However if

119871119909

gt 119871cr119909then the prevailing role in the development of

baroclinic instability plays the vertical wind shear that isthe meridional temperature gradient For example if 120590lowast =

2 times 10minus6m2sdotPaminus2sdotsminus1 and 119906

119879= 75msdotsminus1 then 119871

cr119909asymp 3800 km

5 Concluding Remarks

Geoengineering has appeared as a potential option to reducethe impacts of climate change So far however the effec-tiveness of geoengineering methods is examined outside ofthe scope of optimal control theory and geoengineeringitself is considered as an intentional action to influencenatural climate processes Geophysical cybernetics provides aconceptual and unified theoretical framework for developingand synthesizing the optimal control systems for naturalenvironmental phenomena and processes The applicationof geophysical cybernetics requires a suitable mathematicalmodel of the ECS In mathematical climate models controlactions that manipulate the weather and climate can beexpressed via variations in the model parameters chosen ascontrol variables It is very important that control variablesshould be physically feasible The use of sensitivity theory indynamical systems allows one to determine control variablesthat satisfy this requirement

In this paper bearing in mind the control problem forlarge-scale atmospheric dynamics we considered the atmo-spheric baroclinic instability as the controlled object Withinthe framework of two-layer atmospheric model used in thisstudy there are two fundamental atmospheric parametersthat govern the development of baroclinic instability namelythe static stability and the vertical wind shear induced bythe meridional temperature gradient The influence of smallvariations in these two parameters on the development ofbaroclinic instability has been studied Analytical expressionswere derived for absolute and relative sensitivity coefficientsthat allow one to estimate the absolute and relative contri-bution of variations in the static stability and vertical windshear to changes in the growth rates of unstable baroclinicwaves It was shown that changing the vertical stratification ofthe atmosphere (ie changing the static stability) is the mostappropriate method for controlling the baroclinic instabilityThe influence of meridional temperature gradient on thegrowth rate of unstable waves is less significant Thereforethe vertical wind shear can hardly be regarded as a controlvariable

Let us emphasize that climate manipulation is a mul-tidisciplinary research area that requires consideration notonly of the mathematical aspects but also of the physicalchemical technical ethical and legal aspects and limitationsThe interest in manipulation of the climate and weather willlikely continue to grow which requires the development oftheoretical foundation for the optimal control of the ECSThe

approach outlined in this paper is expected to be applied forthe study of sensitivity of climate and atmospheric models inorder to estimate the hypothetical possibility of weather andclimate optimal control

Competing Interests

The authors declare that they have no competing interests

References

[1] A S Dennis Changing of Weather by Cloud Seeding AcademicPress New York NY USA 1980

[2] M Curic D Janc and V Vuckovic ldquoCloud seeding impact onprecipitation as revealed by cloud-resolving mesoscale modelrdquoMeteorology and Atmospheric Physics vol 95 no 3-4 pp 179ndash193 2007

[3] D L Mitchell and W Finnegan ldquoModification of cirrus cloudsto reduce global warmingrdquo Environmental Research Letters vol4 no 4 Article ID 045102 2009

[4] X Guo D Fu X Li et al ldquoAdvances in cloud physicsand weather modification in Chinardquo Advances in AtmosphericSciences vol 32 no 2 pp 230ndash249 2015

[5] I Colbeck ldquoThe development of fog intensive dispersal oper-ationrdquo in Aerosol Science and Technology Hystory and ReviewsD S Ensor Ed pp 367ndash375 RTI Press Research Triangle ParkNC USA 2011

[6] R N Hoffman ldquoControlling the global weatherrdquo Bulletin ofthe American Meteorological Society vol 83 no 2 pp 241ndash2482002

[7] M Garstang R Bruintjes R Serafin et al ldquoWeather mod-ification finding common groundrdquo Bulletin of the AmericanMeteorological Society vol 86 no 5 pp 647ndash655 2005

[8] J R Fleming Fixing the Sky The Checkered History of Weatherand Climate Control ColumbiaUniversity Press NewYork NYUSA 2010

[9] Geoengineering the Climate Science Governance and Uncer-tainty The Royal Society 2009

[10] T F Stocker D Qin G-K Plattner et al Eds Climate Change2013 The Physical Science Basis Contribution of Working GroupI to the FifthAssessment Report of the Intergovernmental Panel onClimate Change Cambridge University Press Cambridge UK2013

[11] M C MacCracken ldquoOn the possible use of geoengineeringto moderate specific climate change impactsrdquo EnvironmentalResearch Letters vol 4 no 4 Article ID 045107 pp 1ndash14 2009

[12] T Ming R De Richter W Liu and S Caillol ldquoFightingglobal warming by climate engineering is the Earth radiationmanagement and the solar radiation management any optionfor fighting climate changerdquo Renewable and Sustainable EnergyReviews vol 31 pp 792ndash834 2014

[13] R M Yusupov Theoretical Bases of Control of GeophysicalProcesses Ministry of Higher Education of the USSR MoscowRussia 1979

[14] D V Gaskarov V B Kisselev S A Soldatenko V I Strogonovand R M Yusupov An Introduction to Geophysical Cyberneticsand Environmental Monitoring St Petersburg State UniversitySt Petersburg Russia 1998

[15] S Soldatenko and R Yusupov ldquoOn the possible use of geophys-ical cybernetics in climate manipulation (geoengineering) and


weather modificationrdquo WSEAS Transactions on Environmentand Development vol 11 pp 116ndash125 2015

[16] S Soldatenko and R Yusupov ldquoAn optimal control problemformulation for the atmospheric large-scale wave dynamicsrdquoApplied Mathematical Sciences vol 9 no 17ndash20 pp 875ndash8842015

[17] H A Dijkstra Nonlinear Climate Dynamics Cambridge Uni-versity Press New York NY USA 2013

[18] E Rosenwasser andR Yusupov Sensitivity of Automatic ControlSystems CRC Press Boca Raton Fla USA 2000

[19] D G Cacuci Sensitivity and Uncertainty Analysis Volume ITheory CRC Boca Raton Fla USA 2003

[20] J Pedlosky Geophysical Fluid Dynamics Springer New YorkNY USA 1987

[21] M L Salby Fundamental of Atmospheric Physics AcademicPress San Diego Calif USA 1996

[22] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 pp 130ndash140 1963

[23] J R Holton An Introduction to Dynamic Meteorology ElsevierLondon UK 4th edition 2004

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



technologies include for example various engineered solu-tions that remove GHGs directly from the atmosphere usingbig machines or chemical absorbents SRM techniques aimmainly to reflect some percentage of the solar radiation backinto the outer space This can be achieved by changing theplanetary albedo (increasing the reflectiveness of the Earthrsquossurface or clouds) dispersing small particles into the strato-sphere or deploying mirrors in the upper atmosphere toreflect sunlight However all of these technologies introduceuncertainties and unexpected consequences that must beexplored

Let both weather modification and climate engineeringhereafter be referred to as the geoengineering Realization ofgeoengineering projects is a purposeful process that is theprocess imminently connected with a specific objective thatcan be formulated in various ways In this context geoengi-neering is per se the process of controlling the ECS Howevergeoengineering is still considered outside of the scope ofcontrol theory as an intentional action to influence naturalprocessesMeanwhile in the late 1970s a uniformmethodologyfor control geophysical processes including processes occur-ring in the ECS was formulated on the basis of the ideasfrom cybernetics by one of the authors of this paper [13] Inthis monograph the concept of geophysical cybernetics wasintroduced as the new research areawithin the control theoryGeophysical cybernetics explores a self-regulating feedbackcybernetic system in which the ECS is considered as thecontrol object and the role of the controller is given to humanoperators From the standpoint of geophysical cyberneticsclimate and weather manipulation represents an optimalcontrol problem which aims to synthesize the control lawthat ensures the achievement of the desired results that canbe expressed in terms of extremal problem [13 14] In ourprevious publications [14ndash16] an optimal control problemfor the ECS has been conceptually formulated in bothprobabilistic and deterministicmanner Let us emphasize thatthe ECS is a unique natural physical system with a largenumber of specific attributes [17] which makes the controlproblem for this system extremely complex The develop-ment of physically feasible methods to control the ECSrequires the determination of feasible control mechanismsand variables The theory of sensitivity in dynamical systems[18 19] serves as a theoretical instrument for solving thisproblem

This paper presents a short description of the ECS as acontrol system emphasizing its unique physical propertiesrelevant to the control problems of large dynamical systemsIt is also noted that the amount of energy released duringprocesses that drive the ECS is orders of magnitudes greaterthan human capabilities which makes the control of the ECSvia direct interventions very problematic However somephysical and dynamical processes occurring in the ECS areinherently unstable which apparently allows one to imple-ment geoengineering projects using significantly less energyresources in comparisonwith the energy of natural processesThus the study of various types of instabilities in ECS is veryimportant for developing an appropriate control strategyTheinstability of natural physical processes is mathematicallystudied as a problem of finding the necessary conditions for

the growth of the infinitesimal perturbations Since theseconditions are expressed in terms of some model parame-ters by using sensitivity analysis we can first determineparameters that can be considered as controls and secondmake a conclusion on the hypothetical possibility of controlof physical process under consideration To illustrate thisldquoinstability-sensitivityrdquo approach we as an example con-sider the atmospheric baroclinic instability as the controlledobject since this type of hydrodynamic instability plays asubstantial role in the formation of large-scale atmosphericeddies in the extratropical atmosphere and therefore in thegeneral circulations of the atmosphere and climate [20] Toexplore the response of baroclinic instability to geoengi-neering interventions a multilayer geostrophic model of theatmosphere is appliedThe growth rate of baroclinic unstablewaves is taken as an indicator of control manipulations Theinformation obtained via calculated sensitivity coefficientsis very beneficial for assessing the physical feasibility ofmethods of control of the large-scale atmospheric dynam-ics and for designing optimal control systems for climaticprocesses

2 Climate System as a Unique Control System

TheECS is a unique and peculiar natural physical system thatis extremely difficult to control since it possesses a number ofspecific properties including but not limited to the following[14 17 21 22]

(i) The ECS is a complex interactive system with a widevariety of positive and negative feedback mechanisms TheECS consists of the atmosphere ocean sea-ice land surfaceand other bodies of water and includes global carbon cyclechemistry and aerosols These natural subsystems have sub-stantial differences in their physical and chemical propertiesstructure and behavior they can be strongly or weaklycoupled and linked together by means of coupling physicalmechanisms

(ii) Dynamical and physical processes in the ECS occurover a broad spectrum of scales in both space and timeTime scales are varied from seconds (turbulent fluctuations)to dozens of years (climate change and variability) Since theECS is a global system its spatial spectrum of motions coversmolecular to planetary scales It is important that dynamicalprocesses in the atmosphere and ocean are nonlinear andchaotic

(iii) Processes in the ECS oscillate due to bothinternal factors (natural oscillations) and external forcing(forced oscillations) Natural oscillations are due to theinternal instability of ECS with respect to stochasticinfinitesimal disturbances Human impact on the ECS bothintentional and unintentional is considered as externalforcing

The ECS has certainly a number of other specificattributes that make it a unique and complex physical systemOne of the most effective instruments in studying the ECSis mathematicalnumerical modeling Climate mathematicalmodels used in a variety of applications are commonlydeterministic and derived from a set of multidimensional












120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910+ 120596

120597119906

120597119901minus 119891V = minus

120597Φ

120597119909 (1)

120597V120597119905

+ 119906120597V120597119909

+ V120597V120597119910

+ 120596120597V120597119901

+ 119891119906 = minus120597Φ

120597119910 (2)

119877119879

119901= minus

120597Φ

120597119901 (3)

120597119906

120597119909+

120597V120597119910

+120597120596

120597119901= 0 (4)

120597119879

120597119905+ 119906

120597119879

120597119909+ V

120597119879

120597119910minus

119877119879

119892119901(120574119889minus 120574) 120596 = 0 (5)


119889


0+ 120573119910 where





119906 = minus1

1198910

120597Φ

120597119910

V = 0

120596 = 0

120597Φ

120597119901= minus

119877119879

119901

(6)


Pressure (hPa)0

250

500

750

1000

0

1

2

3

4

Level1205960

1205962

1205964

1205951

1205953



120597119906

120597119901=

119877

1198910119901

120597119879

120597119910 (7)


(120597

120597119905+ 119906

120597

120597119909+ V

120597

120597119910)nabla2120595 + 120573

120597120595

120597119901= 1198910

120597120596

120597119901 (8)

(120597

120597119905+ 119906

120597

120597119909+ V

120597

120597119910)

120597120595

120597119901+

120590

1198910

120596 = 0 (9)




120596 = 0 at 119901 = 0 119901 = 1198750 (10)




120597


21205951) + 120573

120597120595

120597119909=

1198910

Δ1199011205962 (11)

120597


21205953) + 120573

120597120595

120597119909= minus

1198910

Δ1199011205962 (12)


120597



120590Δ119901

1198910

1205962= 0 (13)


2 1205951 and 120595



1205951= minus1199061119910 + 120595

1015840

1(119909 119905)

1205953= minus1199063119910 + 120595

1015840

3(119909 119905)

1205962= 1205961015840

2(119909 119905)

(14)


119906119898

=1

2(1199061+ 1199063)

119906119879=

1

2(1199061minus 1199063)

120595119898

=1

2(1205951015840

1+ 1205951015840

3)

120595119879=

1

2(1205951015840

1minus 1205951015840

3)

(15)


tion equations

(120597

120597119905+ 119906119898

120597

120597119909)

1205972120595119898

1205971199092+ 120573

120597120595119898

120597119909+ 119906119879

120597

120597119909(1205972120595119879

1205971199092) = 0

(120597

120597119905+ 119906119898

120597

120597119909)(

1205972120595119879

1205971199092minus 21205822120595119879) + 120573

120597120595119879

120597119909

+ 119906119879

120597

120597119909(1205972120595119879

1205971199092+ 21205822120595119879) = 0

(16)

where 1205822

= 1198912

0[120590(Δ119901)



120595119898(119909 119905) = Ψ

119898119890119894119896(119909minus119888119905)

120595119879(119909 119905) = Ψ

119879119890119894119896(119909minus119888119905)

(17)

where Ψ119898



119888 = 119906119898minus

120573

1198962

1198962+ 1205822

1198962 + 21205822plusmn radic120575 (18)

where

120575 =12057321205824

1198964 (1198962 + 21205822)2+ 1199062

119879

1198962minus 21205822

1198962 + 21205822 (19)



120001000080006000400020000

Wavelength (km)

0

5

10

15

20

25

30

35

40

Ther

mal

win

d (m

middotsminus1)

ci gt 0

ci = 0



120594119896equiv 119896119888119894=

radic100381610038161003816100381612057321205824 + 1199062

1198791198964 (1198964 minus 41205824)

1003816100381610038161003816

119896 (1198962 + 21205822) (20)




119879and




1199062

119879=

12057321205824

1198964 (41205824 minus 1198964) (21)


119909satisfies the



1198964= 21205824plusmn (4120582

8minus

12057321205824

1199062119879

)

12

(22)




119879| gt


119879 then





0=




= 75msminus1 and119906119898




120590and 119878




119878120590equiv

120597120594119896

120597120590= minus

1198961205822

120590 (1198962 + 21205822)2

21199062

1198791198964(1198964+ 21205822) minus 12057321205822

11990621198791198964 (1198964 minus 41205824) + 12057321205824

sdot radic11990621198791198964 (1198964 minus 41205824) + 12057321205824

119878119906119879

equiv120597120594119896

120597119906119879

= minus1199061198791198963(1198962minus 21205822)

sdot

radic100381610038161003816100381612057321205824 + 1199062

1198791198964 (1198964 minus 41205824)

1003816100381610038161003816

12057321205824 + 11990621198791198964 (1198964 minus 41205824)

(23)













119909asymp 3 000 and 119871

(119861)



lowast



(119860)


119878(119861)




1000080006000400020000

Wavelength (km)

minus30

minus20

minus10

0

10

20

30

Wav

e spe

ed (m

sminus1)




minus2

minus4

minus6

minus8

minus10

minus12

minus14

700060005000400030002000

0

Wavelength (km)

Sens

itivi

tyS120590

120590lowast= 10

120590lowast= 15

120590lowast= 20

120590lowast= 25

120590lowast= 30






(119860)



120590lowast and 120575119906 ≪ 119906

lowast


119896


120575120594119896(120575120590) equiv 120594

119896(120590lowast+ 120575120590) minus 120594


120590

1003816100381610038161003816120590=120590lowast (24)

120575120594119896(120575119906) equiv 120594

119896(119906lowast

119879+ 120575119906) minus 120594

119896(119906lowast

119879) asymp 120575119906 times 119878

119906119879

10038161003816100381610038161003816119906119879=119906lowast

119879

(25)





asymp 119871(119860)




119909)



2000 4000 6000 8000

Sens

itivi

tySU119879

020

015

010

005

000

Wavelength (km)

ulowastT = 6

ulowastT = 8

ulowastT = 10

ulowastT = 12

ulowastT = 14





SR 120590SR u119905

|SR120590 |SRu119905

Lcrx

Wavelength (km)6000500040003000

6

5

4

3

2

1

0


120590and 119878

119877




=

75msdotsminus1



lowast




lowast

119879



lowast


119871119909km 5000 4000 3500 3200 3150

120594 dayminus1 037 044 037 021 015119878120590


120575120594 dayminus1 001 002 004 009 013(120575120594120594) times 100 27 45 108 429 867


lowast

119879 120590lowast = 2 times 10



119871119909km 5000 4000 3500 3200 3150

119878119906119879

011 008 007 005 005120575120594 dayminus1 004 003 003 003 002(120575120594120594) times 100 108 68 81 95 123







119878119877

120590= 119878120590

120590

120594119896

119878119877

119906119879= 119878119906119879

119906119879

120594119896

(26)





119909lt 119871


119871119909











Competing Interests


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910+ 120596

120597119906

120597119901minus 119891V = minus

120597Φ

120597119909 (1)

120597V120597119905

+ 119906120597V120597119909

+ V120597V120597119910

+ 120596120597V120597119901

+ 119891119906 = minus120597Φ

120597119910 (2)

119877119879

119901= minus

120597Φ

120597119901 (3)

120597119906

120597119909+

120597V120597119910

+120597120596

120597119901= 0 (4)

120597119879

120597119905+ 119906

120597119879

120597119909+ V

120597119879

120597119910minus

119877119879

119892119901(120574119889minus 120574) 120596 = 0 (5)


119889


0+ 120573119910 where





119906 = minus1

1198910

120597Φ

120597119910

V = 0

120596 = 0

120597Φ

120597119901= minus

119877119879

119901

(6)


Pressure (hPa)0

250

500

750

1000

0

1

2

3

4

Level1205960

1205962

1205964

1205951

1205953



120597119906

120597119901=

119877

1198910119901

120597119879

120597119910 (7)


(120597

120597119905+ 119906

120597

120597119909+ V

120597

120597119910)nabla2120595 + 120573

120597120595

120597119901= 1198910

120597120596

120597119901 (8)

(120597

120597119905+ 119906

120597

120597119909+ V

120597

120597119910)

120597120595

120597119901+

120590

1198910

120596 = 0 (9)




120596 = 0 at 119901 = 0 119901 = 1198750 (10)




120597


21205951) + 120573

120597120595

120597119909=

1198910

Δ1199011205962 (11)

120597


21205953) + 120573

120597120595

120597119909= minus

1198910

Δ1199011205962 (12)


120597



120590Δ119901

1198910

1205962= 0 (13)


2 1205951 and 120595



1205951= minus1199061119910 + 120595

1015840

1(119909 119905)

1205953= minus1199063119910 + 120595

1015840

3(119909 119905)

1205962= 1205961015840

2(119909 119905)

(14)


119906119898

=1

2(1199061+ 1199063)

119906119879=

1

2(1199061minus 1199063)

120595119898

=1

2(1205951015840

1+ 1205951015840

3)

120595119879=

1

2(1205951015840

1minus 1205951015840

3)

(15)


tion equations

(120597

120597119905+ 119906119898

120597

120597119909)

1205972120595119898

1205971199092+ 120573

120597120595119898

120597119909+ 119906119879

120597

120597119909(1205972120595119879

1205971199092) = 0

(120597

120597119905+ 119906119898

120597

120597119909)(

1205972120595119879

1205971199092minus 21205822120595119879) + 120573

120597120595119879

120597119909

+ 119906119879

120597

120597119909(1205972120595119879

1205971199092+ 21205822120595119879) = 0

(16)

where 1205822

= 1198912

0[120590(Δ119901)



120595119898(119909 119905) = Ψ

119898119890119894119896(119909minus119888119905)

120595119879(119909 119905) = Ψ

119879119890119894119896(119909minus119888119905)

(17)

where Ψ119898



119888 = 119906119898minus

120573

1198962

1198962+ 1205822

1198962 + 21205822plusmn radic120575 (18)

where

120575 =12057321205824

1198964 (1198962 + 21205822)2+ 1199062

119879

1198962minus 21205822

1198962 + 21205822 (19)



120001000080006000400020000

Wavelength (km)

0

5

10

15

20

25

30

35

40

Ther

mal

win

d (m

middotsminus1)

ci gt 0

ci = 0



120594119896equiv 119896119888119894=

radic100381610038161003816100381612057321205824 + 1199062

1198791198964 (1198964 minus 41205824)

1003816100381610038161003816

119896 (1198962 + 21205822) (20)




119879and




1199062

119879=

12057321205824

1198964 (41205824 minus 1198964) (21)


119909satisfies the



1198964= 21205824plusmn (4120582

8minus

12057321205824

1199062119879

)

12

(22)




119879| gt


119879 then





0=




= 75msminus1 and119906119898




120590and 119878




119878120590equiv

120597120594119896

120597120590= minus

1198961205822

120590 (1198962 + 21205822)2

21199062

1198791198964(1198964+ 21205822) minus 12057321205822

11990621198791198964 (1198964 minus 41205824) + 12057321205824

sdot radic11990621198791198964 (1198964 minus 41205824) + 12057321205824

119878119906119879

equiv120597120594119896

120597119906119879

= minus1199061198791198963(1198962minus 21205822)

sdot

radic100381610038161003816100381612057321205824 + 1199062

1198791198964 (1198964 minus 41205824)

1003816100381610038161003816

12057321205824 + 11990621198791198964 (1198964 minus 41205824)

(23)













119909asymp 3 000 and 119871

(119861)



lowast



(119860)


119878(119861)




1000080006000400020000

Wavelength (km)

minus30

minus20

minus10

0

10

20

30

Wav

e spe

ed (m

sminus1)




minus2

minus4

minus6

minus8

minus10

minus12

minus14

700060005000400030002000

0

Wavelength (km)

Sens

itivi

tyS120590

120590lowast= 10

120590lowast= 15

120590lowast= 20

120590lowast= 25

120590lowast= 30






(119860)



120590lowast and 120575119906 ≪ 119906

lowast


119896


120575120594119896(120575120590) equiv 120594

119896(120590lowast+ 120575120590) minus 120594


120590

1003816100381610038161003816120590=120590lowast (24)

120575120594119896(120575119906) equiv 120594

119896(119906lowast

119879+ 120575119906) minus 120594

119896(119906lowast

119879) asymp 120575119906 times 119878

119906119879

10038161003816100381610038161003816119906119879=119906lowast

119879

(25)





asymp 119871(119860)




119909)



2000 4000 6000 8000

Sens

itivi

tySU119879

020

015

010

005

000

Wavelength (km)

ulowastT = 6

ulowastT = 8

ulowastT = 10

ulowastT = 12

ulowastT = 14





SR 120590SR u119905

|SR120590 |SRu119905

Lcrx

Wavelength (km)6000500040003000

6

5

4

3

2

1

0


120590and 119878

119877




=

75msdotsminus1



lowast




lowast

119879



lowast


119871119909km 5000 4000 3500 3200 3150

120594 dayminus1 037 044 037 021 015119878120590


120575120594 dayminus1 001 002 004 009 013(120575120594120594) times 100 27 45 108 429 867


lowast

119879 120590lowast = 2 times 10



119871119909km 5000 4000 3500 3200 3150

119878119906119879

011 008 007 005 005120575120594 dayminus1 004 003 003 003 002(120575120594120594) times 100 108 68 81 95 123







119878119877

120590= 119878120590

120590

120594119896

119878119877

119906119879= 119878119906119879

119906119879

120594119896

(26)





119909lt 119871


119871119909











Competing Interests


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Pressure (hPa)0

250

500

750

1000

0

1

2

3

4

Level1205960

1205962

1205964

1205951

1205953



120597119906

120597119901=

119877

1198910119901

120597119879

120597119910 (7)


(120597

120597119905+ 119906

120597

120597119909+ V

120597

120597119910)nabla2120595 + 120573

120597120595

120597119901= 1198910

120597120596

120597119901 (8)

(120597

120597119905+ 119906

120597

120597119909+ V

120597

120597119910)

120597120595

120597119901+

120590

1198910

120596 = 0 (9)




120596 = 0 at 119901 = 0 119901 = 1198750 (10)




120597


21205951) + 120573

120597120595

120597119909=

1198910

Δ1199011205962 (11)

120597


21205953) + 120573

120597120595

120597119909= minus

1198910

Δ1199011205962 (12)


120597



120590Δ119901

1198910

1205962= 0 (13)


2 1205951 and 120595



1205951= minus1199061119910 + 120595

1015840

1(119909 119905)

1205953= minus1199063119910 + 120595

1015840

3(119909 119905)

1205962= 1205961015840

2(119909 119905)

(14)


119906119898

=1

2(1199061+ 1199063)

119906119879=

1

2(1199061minus 1199063)

120595119898

=1

2(1205951015840

1+ 1205951015840

3)

120595119879=

1

2(1205951015840

1minus 1205951015840

3)

(15)


tion equations

(120597

120597119905+ 119906119898

120597

120597119909)

1205972120595119898

1205971199092+ 120573

120597120595119898

120597119909+ 119906119879

120597

120597119909(1205972120595119879

1205971199092) = 0

(120597

120597119905+ 119906119898

120597

120597119909)(

1205972120595119879

1205971199092minus 21205822120595119879) + 120573

120597120595119879

120597119909

+ 119906119879

120597

120597119909(1205972120595119879

1205971199092+ 21205822120595119879) = 0

(16)

where 1205822

= 1198912

0[120590(Δ119901)



120595119898(119909 119905) = Ψ

119898119890119894119896(119909minus119888119905)

120595119879(119909 119905) = Ψ

119879119890119894119896(119909minus119888119905)

(17)

where Ψ119898



119888 = 119906119898minus

120573

1198962

1198962+ 1205822

1198962 + 21205822plusmn radic120575 (18)

where

120575 =12057321205824

1198964 (1198962 + 21205822)2+ 1199062

119879

1198962minus 21205822

1198962 + 21205822 (19)



120001000080006000400020000

Wavelength (km)

0

5

10

15

20

25

30

35

40

Ther

mal

win

d (m

middotsminus1)

ci gt 0

ci = 0



120594119896equiv 119896119888119894=

radic100381610038161003816100381612057321205824 + 1199062

1198791198964 (1198964 minus 41205824)

1003816100381610038161003816

119896 (1198962 + 21205822) (20)




119879and




1199062

119879=

12057321205824

1198964 (41205824 minus 1198964) (21)


119909satisfies the



1198964= 21205824plusmn (4120582

8minus

12057321205824

1199062119879

)

12

(22)




119879| gt


119879 then





0=




= 75msminus1 and119906119898




120590and 119878




119878120590equiv

120597120594119896

120597120590= minus

1198961205822

120590 (1198962 + 21205822)2

21199062

1198791198964(1198964+ 21205822) minus 12057321205822

11990621198791198964 (1198964 minus 41205824) + 12057321205824

sdot radic11990621198791198964 (1198964 minus 41205824) + 12057321205824

119878119906119879

equiv120597120594119896

120597119906119879

= minus1199061198791198963(1198962minus 21205822)

sdot

radic100381610038161003816100381612057321205824 + 1199062

1198791198964 (1198964 minus 41205824)

1003816100381610038161003816

12057321205824 + 11990621198791198964 (1198964 minus 41205824)

(23)













119909asymp 3 000 and 119871

(119861)



lowast



(119860)


119878(119861)




1000080006000400020000

Wavelength (km)

minus30

minus20

minus10

0

10

20

30

Wav

e spe

ed (m

sminus1)




minus2

minus4

minus6

minus8

minus10

minus12

minus14

700060005000400030002000

0

Wavelength (km)

Sens

itivi

tyS120590

120590lowast= 10

120590lowast= 15

120590lowast= 20

120590lowast= 25

120590lowast= 30






(119860)



120590lowast and 120575119906 ≪ 119906

lowast


119896


120575120594119896(120575120590) equiv 120594

119896(120590lowast+ 120575120590) minus 120594


120590

1003816100381610038161003816120590=120590lowast (24)

120575120594119896(120575119906) equiv 120594

119896(119906lowast

119879+ 120575119906) minus 120594

119896(119906lowast

119879) asymp 120575119906 times 119878

119906119879

10038161003816100381610038161003816119906119879=119906lowast

119879

(25)





asymp 119871(119860)




119909)



2000 4000 6000 8000

Sens

itivi

tySU119879

020

015

010

005

000

Wavelength (km)

ulowastT = 6

ulowastT = 8

ulowastT = 10

ulowastT = 12

ulowastT = 14





SR 120590SR u119905

|SR120590 |SRu119905

Lcrx

Wavelength (km)6000500040003000

6

5

4

3

2

1

0


120590and 119878

119877




=

75msdotsminus1



lowast




lowast

119879



lowast


119871119909km 5000 4000 3500 3200 3150

120594 dayminus1 037 044 037 021 015119878120590


120575120594 dayminus1 001 002 004 009 013(120575120594120594) times 100 27 45 108 429 867


lowast

119879 120590lowast = 2 times 10



119871119909km 5000 4000 3500 3200 3150

119878119906119879

011 008 007 005 005120575120594 dayminus1 004 003 003 003 002(120575120594120594) times 100 108 68 81 95 123







119878119877

120590= 119878120590

120590

120594119896

119878119877

119906119879= 119878119906119879

119906119879

120594119896

(26)





119909lt 119871


119871119909











Competing Interests


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120001000080006000400020000

Wavelength (km)

0

5

10

15

20

25

30

35

40

Ther

mal

win

d (m

middotsminus1)

ci gt 0

ci = 0



120594119896equiv 119896119888119894=

radic100381610038161003816100381612057321205824 + 1199062

1198791198964 (1198964 minus 41205824)

1003816100381610038161003816

119896 (1198962 + 21205822) (20)




119879and




1199062

119879=

12057321205824

1198964 (41205824 minus 1198964) (21)


119909satisfies the



1198964= 21205824plusmn (4120582

8minus

12057321205824

1199062119879

)

12

(22)




119879| gt


119879 then





0=




= 75msminus1 and119906119898




120590and 119878




119878120590equiv

120597120594119896

120597120590= minus

1198961205822

120590 (1198962 + 21205822)2

21199062

1198791198964(1198964+ 21205822) minus 12057321205822

11990621198791198964 (1198964 minus 41205824) + 12057321205824

sdot radic11990621198791198964 (1198964 minus 41205824) + 12057321205824

119878119906119879

equiv120597120594119896

120597119906119879

= minus1199061198791198963(1198962minus 21205822)

sdot

radic100381610038161003816100381612057321205824 + 1199062

1198791198964 (1198964 minus 41205824)

1003816100381610038161003816

12057321205824 + 11990621198791198964 (1198964 minus 41205824)

(23)













119909asymp 3 000 and 119871

(119861)



lowast



(119860)


119878(119861)




1000080006000400020000

Wavelength (km)

minus30

minus20

minus10

0

10

20

30

Wav

e spe

ed (m

sminus1)




minus2

minus4

minus6

minus8

minus10

minus12

minus14

700060005000400030002000

0

Wavelength (km)

Sens

itivi

tyS120590

120590lowast= 10

120590lowast= 15

120590lowast= 20

120590lowast= 25

120590lowast= 30






(119860)



120590lowast and 120575119906 ≪ 119906

lowast


119896


120575120594119896(120575120590) equiv 120594

119896(120590lowast+ 120575120590) minus 120594


120590

1003816100381610038161003816120590=120590lowast (24)

120575120594119896(120575119906) equiv 120594

119896(119906lowast

119879+ 120575119906) minus 120594

119896(119906lowast

119879) asymp 120575119906 times 119878

119906119879

10038161003816100381610038161003816119906119879=119906lowast

119879

(25)





asymp 119871(119860)




119909)



2000 4000 6000 8000

Sens

itivi

tySU119879

020

015

010

005

000

Wavelength (km)

ulowastT = 6

ulowastT = 8

ulowastT = 10

ulowastT = 12

ulowastT = 14





SR 120590SR u119905

|SR120590 |SRu119905

Lcrx

Wavelength (km)6000500040003000

6

5

4

3

2

1

0


120590and 119878

119877




=

75msdotsminus1



lowast




lowast

119879



lowast


119871119909km 5000 4000 3500 3200 3150

120594 dayminus1 037 044 037 021 015119878120590


120575120594 dayminus1 001 002 004 009 013(120575120594120594) times 100 27 45 108 429 867


lowast

119879 120590lowast = 2 times 10



119871119909km 5000 4000 3500 3200 3150

119878119906119879

011 008 007 005 005120575120594 dayminus1 004 003 003 003 002(120575120594120594) times 100 108 68 81 95 123







119878119877

120590= 119878120590

120590

120594119896

119878119877

119906119879= 119878119906119879

119906119879

120594119896

(26)





119909lt 119871


119871119909











Competing Interests


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1000080006000400020000

Wavelength (km)

minus30

minus20

minus10

0

10

20

30

Wav

e spe

ed (m

sminus1)




minus2

minus4

minus6

minus8

minus10

minus12

minus14

700060005000400030002000

0

Wavelength (km)

Sens

itivi

tyS120590

120590lowast= 10

120590lowast= 15

120590lowast= 20

120590lowast= 25

120590lowast= 30






(119860)



120590lowast and 120575119906 ≪ 119906

lowast


119896


120575120594119896(120575120590) equiv 120594

119896(120590lowast+ 120575120590) minus 120594


120590

1003816100381610038161003816120590=120590lowast (24)

120575120594119896(120575119906) equiv 120594

119896(119906lowast

119879+ 120575119906) minus 120594

119896(119906lowast

119879) asymp 120575119906 times 119878

119906119879

10038161003816100381610038161003816119906119879=119906lowast

119879

(25)





asymp 119871(119860)




119909)



2000 4000 6000 8000

Sens

itivi

tySU119879

020

015

010

005

000

Wavelength (km)

ulowastT = 6

ulowastT = 8

ulowastT = 10

ulowastT = 12

ulowastT = 14





SR 120590SR u119905

|SR120590 |SRu119905

Lcrx

Wavelength (km)6000500040003000

6

5

4

3

2

1

0


120590and 119878

119877




=

75msdotsminus1



lowast




lowast

119879



lowast


119871119909km 5000 4000 3500 3200 3150

120594 dayminus1 037 044 037 021 015119878120590


120575120594 dayminus1 001 002 004 009 013(120575120594120594) times 100 27 45 108 429 867


lowast

119879 120590lowast = 2 times 10



119871119909km 5000 4000 3500 3200 3150

119878119906119879

011 008 007 005 005120575120594 dayminus1 004 003 003 003 002(120575120594120594) times 100 108 68 81 95 123







119878119877

120590= 119878120590

120590

120594119896

119878119877

119906119879= 119878119906119879

119906119879

120594119896

(26)





119909lt 119871


119871119909











Competing Interests


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2000 4000 6000 8000

Sens

itivi

tySU119879

020

015

010

005

000

Wavelength (km)

ulowastT = 6

ulowastT = 8

ulowastT = 10

ulowastT = 12

ulowastT = 14





SR 120590SR u119905

|SR120590 |SRu119905

Lcrx

Wavelength (km)6000500040003000

6

5

4

3

2

1

0


120590and 119878

119877




=

75msdotsminus1



lowast




lowast

119879



lowast


119871119909km 5000 4000 3500 3200 3150

120594 dayminus1 037 044 037 021 015119878120590


120575120594 dayminus1 001 002 004 009 013(120575120594120594) times 100 27 45 108 429 867


lowast

119879 120590lowast = 2 times 10



119871119909km 5000 4000 3500 3200 3150

119878119906119879

011 008 007 005 005120575120594 dayminus1 004 003 003 003 002(120575120594120594) times 100 108 68 81 95 123







119878119877

120590= 119878120590

120590

120594119896

119878119877

119906119879= 119878119906119879

119906119879

120594119896

(26)





119909lt 119871


119871119909











Competing Interests


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119909lt 119871


119871119909











Competing Interests


References




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article the determination of feasible control...

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