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A cellulose pulp continuous digester (e.g. for producing kraft pulp) is operated so that it has the advantages of a hydraulic digester yet has a gas-filled zone over the liquid level. A slurry of chips and cooking liquor is introduced into the top of the digester vessel through an inverted top separator. A liquid level is established below the inverted top separator, and a chips level is established below the inverted top separator (above or below the liquid level). A gas-filled zone above the liquid level includes compressed gas, and is at a temperature of less than 160 (preferably less than 120) DEG C. and at a pressure of between 50-200 (preferably 80-150) psig. The digester may have a first diameter portion (about 3-5 meters) at the top and a second diameter portion (at least about seven meters) below the top, with the inverted separator in the first diameter portion, the liquid level in the first diameter portion, and the chips level in the second diameter portion. A shoulder is provided between the first and second diameter portions and a withdrawal screen is preferably mounted just below the shoulder in the second diameter portion. A plurality of different liquor flows may be extracted from the inverted top separator.