MOST OF THE BARYONS IN A GALAXY CLUSTER RESIDE IN A HOT (10-100 MILLION K) AND TENUOUS GASEOUS ATMOSPHERE CONFINED BY THE GRAVITATIONAL POTENTIAL OF THE CLUSTER'S DARK MATTER HALO. UNDERSTANDING THE MICROPHYSICS OF THIS INTRACLUSTER MEDIUM (ICM) PARTICULARLY THE TRANSPORT PROCESSES SUCH AS THERMAL CONDUCTION AND VISCOSITY IS IMPORTANT TO ANY UNDERSTANDING OF THE THERMODYNAMIC STATE OF ICM ATMOSPHERES. FOR EXAMPLE THE CURRENT PARADIGM IS THAT RADIATIVE LOSSES IN THE ICM CORE ARE OFFSET BY ENERGY FROM A CENTRAL JETTED ACTIVE GALACTIC NUCLEUS (AGN) PREVENTING A COOLING CATASTROPHE IN THE CLUSTER CORE. HOWEVER THE MECHANISM BY WHICH THE JET-INJECTED ENERGY IS THERMALIZED IN THE ICM IS HIGHLY UNCERTAIN - THE DISSIPATION OF WAVES OR TURBULENCE BY THERMAL CONDUCTION OR PLASMA VISCOSITY IS A LEADING CONTENDER. A KNOWLEDGE OF THERMAL CONDUCTION IN THE ICM IS ALSO IMPORTANT FOR ANY ATTEMPTS TO UNDERSTAND THE GLOBAL TEMPERATURE PROFILES OF CLUSTERS WITH CONSEQUENCES FOR E.G. COSMOLOGICAL STUDIES BASED ON OBSERVATIONS OF THE SUNYAEV-ZELDOVICH (SZ) EFFECT. THE BASIC PHYSICS OF THERMAL CONDUCTION IN THE ICM IS VERY POORLY UNDERSTOOD HOWEVER LEADING TO A HUGE UNCERTAINTY IN THE RELEVANT COEFFICIENTS. THE ICM RESIDES IN A POORLY STUDIED REGIME OF PLASMA PHYSICS - IT IS A HIGHLY MAGNETIZED (GYRORADII<<PARTICLE MEAN FREE PATH) HIGH-BETA (THERMAL PRESSURE>>MAGNETIC PRESSURE) AND WEAKLY COLLISIONAL (MEAN-FREE PATH ONLY MODERATELY LESS THAN GLOBAL SCALE LENGTHS) PLASMA. THERMAL CONDUCTION WILL BE STRONGLY SUPPRESSED PERPENDICULAR TO MAGNETIC FIELDS LINES. BUT EVEN ALONG FIELD LINES THE GROWTH OF SMALL SCALE AND FAST KINETIC INSTABILITIES MAY STRONGLY SUPPRESS THERMAL CONDUCTION. HENCE THE USUAL ASSUMPTION THAT CONDUCTION ALONG THE FIELD HAS ITS CLASSICAL SPITZER VALUE HAS A SHAKY THEORETICAL BASIS AND MAY WELL BE WILDLY INACCURATE. IN THIS PROPOSAL WE USE ANALYTICAL THEORY AND COMPUTER MODELS TO EXPLORE THERMAL CONDUCTION IN ICM-LIKE PLASMAS. RECENTLY WE HAVE FOUND THAT A STRONG HEAT-FLUX WILL DRIVE A POWERFUL WHISTLER-WAVE INSTABILITY AND PROVIDED WE TREAT THE PROBLEM IN MORE THAN 1D SO THAT OBLIQUE MODES ARE CAPTURED THESE WAVES EFFICIENTLY SCATTER ELECTRONS THEREBY SHUTTING DOWN THE HEAT-FLUX. OUR PROPOSED WORK BUILDS ON THESE FINDINGS WITH THE GOAL OF CHARACTERIZING THE MACROSCOPIC EFFECTIVE THERMAL CONDUCTION IN A FORM THAT CAN BE INCLUDED IN FLUID (MAGNETOHYDRODYNAMIC; MHD) MODELS OF THE ICM. WE WILL 1) CONDUCT AN EXTENDED LINEAR ANALYSIS OF THE HEAT-FLUX WHISTLER INSTABILITY EXPLORING THE INTERACTION OF THE HEAT FLUX AND THE PRESSURE ANISOTROPIES THAT WOULD RESULT FROM BULK MOTIONS OF THE ICM. WE WILL MAP THE STABLE/UNSTABLE REGIONS AS A FUNCTION OF HEAT-FLUX PRESSURE ANISOTROPY AND PLASMA-BETA. 2) PERFORM PARTICLE-IN-CELL (PIC) SIMULATIONS TO EXPLORE THE NON-LINEAR SATURATION OF THE HEAT-FLUX WHISTLER INSTABILITY AS A FUNCTION OF THE PLASMA-BETA AND HEAT-FLUX EXTENDING THE CURRENT WORK (I.E. VERY STRONG FLUXES) DOWN TO THE MODEST HEAT-FLUXES FOUND IN THE REAL ICM. KEY IS WHETHER OVERLAPPING WAVE-PARTICLE RESONANCES THAT ARE SO EFFICIENT AT KILLING THE CONDUCTION WITH STRONG HEAT-FLUXES STILL OPERATE WHEN THE DRIVING HEAT-FLUX IS WEAK. 3) DEVELOP A NEW COMPUTATIONAL/PIC MODEL THAT IN CONTRAST TO CURRENT WORK SUSTAINS A TEMPERATURE GRADIENT ACROSS THE DOMAIN THEREBY ALLOWING US TO DIRECTLY MEASURE THE RELATIONSHIP BETWEEN TEMPERATURE GRADIENT AND HEAT FLUX. 4) BUILD A NEW THERMAL CONDUCTION MODEL ALLOWING THE HEAT FLUX TO HAVE A NON-LINEAR DEPENDENCE ON TEMPERATURE GRADIENT AND PLASMA-BETA. WE WILL DEVELOP THERMAL CONDUCTION ALGORITHMS THAT CAN BE USED IN PUBLIC MHD E.G. PLUTO OR FLASH. THIS WORK WILL PROVIDE THE CRUCIAL BRIDGE BETWEEN THE GLOBAL/MHD MODELS OF ICM ATMOSPHERES AND THE MICROPHYSICS THAT DICTATES THE TRANSPORT PROCESSES. IT WILL INFORM THE NEXT GENERATION OF CLUSTER MODELS USED TO INTERPRET DATA FROM NASA'S FLEET OF X-RAY OBSERVATORIES.