Energy Conserving Truncations in Thermal Convection

Abstract

.28> Ø ; Tg = r 2 T + @ Ø @x ; (1) where Ø is the stream function (with v = (\Gamma@ z Ø ; @ x Ø )), T is the temperature deviation from a linear profile, is the kinematic viscosity, and is the thermal conductivity. The Poisson bracket is defined as fA ; Bg j @ x A @ z B \Gamma @ z A @ x B. In the dissipationless limit, where and are set equal to zero, (1) conserves the total energy: E = K + U = 1 2 D (r Ø ) 2 E \Gamma h zT i ; (2) where the angle brackets denote an a