A burner apparatus for combining oxygen and fuel to produce a mixture to be burned, the burner apparatus comprising: an oxygen-delivery system comprising an oxygen-receiving plenum adapted to receive pressurized oxygen from an oxygen source and a primary oxygen-supply housing formed to include a primary-oxygen chamber, an oxygen-discharge outlet opening into the primary-oxygen chamber, a boundary wall arranged to form a boundary of the primary-oxygen chamber and to include several constant-flow oxygen-inlet orifices (R1, R2, C1, C2, C3, C4, L1, L2) arranged to communicate pressurized oxygen constantly from the oxygen-receiving plenum into the primary-oxygen chamber of the primary oxygen-supply housing to establish primary oxygen in the primary oxygen-supply housing, and a vertically extending oxygen-diffuser screen arranged to lie between the several constant-flow oxygen-inlet orifices (R1, R2, C1, C2, C3, C4, L1, L2) and a downstream wall to partition the primary-oxygen chamber into an upstream section and a downstream section, a fuel-discharge nozzle arranged to lie partly in the primary-oxygen chamber and extend through the oxygen-discharge outlet to reach a first-stage combustion zone located outside of the primary-oxygen chamber and configured to conduct fuel received from a fuel source to the first-stage combustion zone to mix with primary oxygen exiting the primary-oxygen chamber through the oxygen-discharge outlet to establish a combustible oxygen-fuel mixture in the first-stage combustion zone that is ignitable to produce a flame having a root lying in the first-stage combustion zone and a tip lying in a second-stage combustion zone downstream from the first-stage combustion zone, wherein the oxygen-delivery system further comprises a staged-oxygen conduit coupled in fluid communication to the oxygen-receiving plenum and configured to conduct staged oxygen discharged under pressure from the oxygen-receiving plenum to the second-stage combustion zone without passing through the primary-oxygen chamber, and a variable oxygen-diversion system comprising means for diverting a variable amount of pressurized oxygen extant in the oxygen-receiving plenum to flow from the oxygen-receiving plenum into the primary-oxygen chamber without flowing through the constant-flow oxygen-inlet orifices (R1, R2, C1, C2, C3, C4, L1, L2) to supplement pressurized oxygen admitted into the primary-oxygen chamber through the constant-flow oxygen-inlet orifices (R1, R2, C1, C2, C3, C4, L1, L2) so that more primary oxygen is provided in the primary-oxygen chamber and delivered through the oxygen-discharge outlet formed in the primary oxygen-supply housing to the first-stage combustion zone and proportionally less staged oxygen is delivered through the staged-oxygen conduit to the second-stage combustion zone. an oxygen-delivery system comprising an oxygen-receiving plenum adapted to receive pressurized oxygen from an oxygen source and a primary oxygen-supply housing formed to include a primary-oxygen chamber, an oxygen-discharge outlet opening into the primary-oxygen chamber, a boundary wall arranged to form a boundary of the primary-oxygen chamber and to include several constant-flow oxygen-inlet orifices (R1, R2, C1, C2, C3, C4, L1, L2) arranged to communicate pressurized oxygen constantly from the oxygen-receiving plenum into the primary-oxygen chamber of the primary oxygen-supply housing to establish primary oxygen in the primary oxygen-supply housing, and a vertically extending oxygen-diffuser screen arranged to lie between the several constant-flow oxygen-inlet orifices (R1, R2, C1, C2, C3, C4, L1, L2) and a downstream wall to partition the primary-oxygen chamber into an upstream section and a downstream section, a fuel-discharge nozzle arranged to lie partly in the primary-oxygen chamber and extend through the oxygen-discharge outlet to reach a first-stage combustion zone located outside of the primary-oxygen chamber and configured to conduct fuel received from a fuel source to the first-stage combustion zone to mix with primary oxygen exiting the primary-oxygen chamber through the oxygen-discharge outlet to establish a combustible oxygen-fuel mixture in the first-stage combustion zone that is ignitable to produce a flame having a root lying in the first-stage combustion zone and a tip lying in a second-stage combustion zone downstream from the first-stage combustion zone, wherein the oxygen-delivery system further comprises a staged-oxygen conduit coupled in fluid communication to the oxygen-receiving plenum and configured to conduct staged oxygen discharged under pressure from the oxygen-receiving plenum to the second-stage combustion zone without passing through the primary-oxygen chamber, and a variable oxygen-diversion system comprising means for diverting a variable amount of pressurized oxygen extant in the oxygen-receiving plenum to flow from the oxygen-receiving plenum into the primary-oxygen chamber without flowing through the constant-flow oxygen-inlet orifices (R1, R2, C1, C2, C3, C4, L1, L2) to supplement pressurized oxygen admitted into the primary-oxygen chamber through the constant-flow oxygen-inlet orifices (R1, R2, C1, C2, C3, C4, L1, L2) so that more primary oxygen is provided in the primary-oxygen chamber and delivered through the oxygen-discharge outlet formed in the primary oxygen-supply housing to the first-stage combustion zone and proportionally less staged oxygen is delivered through the staged-oxygen conduit to the second-stage combustion zone.